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Mathematics LibreTexts

5-2. Unit Circle: Sine and Cosine Functions

Unit Circle: Sine and Cosine Functions In this section, you will: Find function values for the sine and cosine of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>30°</mn><mtext> or </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 6 ), 45° or ( π 4 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> 60° or ( π 3 ). Identify the domain and range of sine and cosine functions. Use reference angles to evaluate trigonometric functions.

<figure class="small" id="Figure_05_02_001" style="color: rgb(0, 0, 0); font-family: 'Times New Roman'; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 1; word-spacing: 0px; -webkit-text-stroke-width: 0px;"> <figcaption>The Singapore Flyer is the world’s tallest Ferris wheel. (credit: “Vibin JK”/Flickr)</figcaption> </figure>

Looking for a thrill? Then consider a ride on the Singapore Flyer, the world’s tallest Ferris wheel. Located in Singapore, the Ferris wheel soars to a height of 541 feet—a little more than a tenth of a mile! Described as an observation wheel, riders enjoy spectacular views as they travel from the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. Then we can discuss circular motion in terms of the coordinate pairs. Finding Function Values for the Sine and Cosine To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1, as shown in [link]. The angle (in radians) that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>intercepts forms an arc of length<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>s</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Using the formula<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>s</mi><mo>=</mo><mi>r</mi><mi>t</mi><mo>,</mo></mrow></annotation-xml></semantics></math> and knowing that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo></mrow></annotation-xml></semantics></math> we see that for a unit circle,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>s</mi><mo>=</mo><mi>t</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> Recall that the x- and y-axes divide the coordinate plane into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV. For any angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>,</mo></mrow></annotation-xml></semantics></math> we can label the intersection of the terminal side and the unit circle as by its coordinates,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ). The coordinates<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext>  </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>will be the outputs of the trigonometric functions<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>,</mo></mrow></annotation-xml></semantics></math> respectively. This means<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>

<figure class="small" id="Figure_05_02_002"> <figcaption>Unit circle where the central angle is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> radians</figcaption> </figure>

Unit Circle A unit circle has a center at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and radius<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>. In a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>1.</mn></mrow></annotation-xml></semantics></math> Let<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) be the endpoint on the unit circle of an arc of arc length <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>s</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>The<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) coordinates of this point can be described as functions of the angle. Defining Sine and Cosine Functions Now that we have our unit circle labeled, we can learn how the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) coordinates relate to the arc length and angle. The sine function relates a real number<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to the y-coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>equals the y-value of the endpoint on the unit circle of an arc of length<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> In [link], the sine is equal to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Like all functions, the sine function has an input and an output. Its input is the measure of the angle; its output is the y-coordinate of the corresponding point on the unit circle. The cosine function of an angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>equals the x-value of the endpoint on the unit circle of an arc of length<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>In [link], the cosine is equal to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>

<figure class="small" id="Figure_05_02_003"> </figure>

Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi></mrow></annotation-xml></semantics></math> is the same as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mi>t</mi></mrow></annotation-xml></semantics></math> is the same as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math> Likewise, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> cos 2 t is a commonly used shorthand notation for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 2 . Be aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra parentheses when entering calculations into a calculator or computer. Sine and Cosine Functions If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is a real number and a point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) on the unit circle corresponds to an angle of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>,</mo></mrow></annotation-xml></semantics></math> then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mi>x</mi></mrow></annotation-xml></semantics></math> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mi>y</mi></mrow></annotation-xml></semantics></math> Given a point P<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>on the unit circle corresponding to an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>,</mo></mrow></annotation-xml></semantics></math> find the sine and cosine. The sine of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>t</mi></mrow></annotation-xml></semantics></math> is equal to the y-coordinate of point <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>P</mi><mo>:</mo><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mi>y</mi><mo>.</mo></mrow></annotation-xml></semantics></math> The cosine of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>t</mi></mrow></annotation-xml></semantics></math> is equal to the x-coordinate of point <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>P</mi><mo>:</mo><mo> </mo><mtext>cos</mtext><mtext> </mtext><mi>t</mi><mo>=</mo><mi>x</mi><mo>.</mo></mrow></annotation-xml></semantics></math> Finding Function Values for Sine and Cosine Point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>  </mtext><mi>P</mi><mtext>  </mtext></mrow></annotation-xml></semantics></math>is a point on the unit circle corresponding to an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>  </mtext><mi>t</mi><mo>,</mo></mrow></annotation-xml></semantics></math> as shown in [link]. Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>  </mtext><mi>cos</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>

<figure class="small" id="Figure_05_02_004"> </figure>

We know that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is the x-coordinate of the corresponding point on the unit circle and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>t</mi><mtext>  </mtext></mrow></annotation-xml></semantics></math>is the y-coordinate of the corresponding point on the unit circle. So: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mtable columnalign="left"><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mi>x</mi><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mtd></mtr></mtable></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 2 y=sin t= 3 2 A certain angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>corresponds to a point on the unit circle at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> − 2 2 , 2 2 ) as shown in [link]. Find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mi>t</mi></mrow></annotation-xml></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<figure class="small" id="Figure_05_02_005"> </figure> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math>

2 2 ,sin(t)= 2 2 Finding Sines and Cosines of Angles on an Axis For quadrantral angles, the corresponding point on the unit circle falls on the x- or y-axis. In that case, we can easily calculate cosine and sine from the values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> Calculating Sines and Cosines along an Axis Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mn>90°</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mo stretchy="false">(</mo><mn>90°</mn><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> Moving<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>90°</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>counterclockwise around the unit circle from the positive x-axis brings us to the top of the circle, where the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>coordinates are (0, 1), as shown in [link].

<figure class="small" id="Figure_05_02_006"> </figure>

Using our definitions of cosine and sine, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mi>cos</mi><mo stretchy="false">(</mo><mn>90°</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mtd></mtr></mtable></annotation-xml></semantics></math> y=sin t=sin(90°)=1 The cosine of 90° is 0; the sine of 90° is 1. Find cosine and sine of the angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>π</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn><mo>,</mo></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math> The Pythagorean Identity Now that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the equation for the unit circle is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> x 2 + y 2 =1.  Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>,</mo></mrow></annotation-xml></semantics></math> we can substitute for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to get<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext></mrow></annotation-xml></semantics></math> cos 2 t+ sin 2 t=1. This equation,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> cos 2 t+ sin 2 t=1, is known as the Pythagorean Identity. See [link].

<figure class="small" id="Figure_05_02_007"> </figure>

We can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or vice versa. However, because the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct sign. If we know the quadrant where the angle is, we can easily choose the correct solution. Pythagorean Identity The Pythagorean Identity states that, for any real number<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>,</mo></mrow></annotation-xml></semantics></math> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mi>cos</mi></mrow></msup></mrow></annotation-xml></semantics></math> 2  t+ sin 2  t=1  Given the sine of some angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext>  </mtext></mrow></annotation-xml></semantics></math>and its quadrant location, find the cosine of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> Substitute the known value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t ) into the Pythagorean Identity. Solve for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t ).  Choose the solution with the appropriate sign for the x-values in the quadrant where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>  </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is located. Finding a Cosine from a Sine or a Sine from a Cosine If <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 7  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is in the second quadrant, find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> If we drop a vertical line from the point on the unit circle corresponding to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>  </mtext><mi>t</mi><mo>,</mo></mrow></annotation-xml></semantics></math> we create a right triangle, from which we can see that the Pythagorean Identity is simply one case of the Pythagorean Theorem. See [link].

<figure class="small" id="Figure_05_02_008"> </figure>

Substituting the known value for sine into the Pythagorean Identity, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mi>cos</mi></mrow></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 (t)+ sin 2 (t)=1            cos 2 (t)+ 9 49 =1                           cos 2 (t)= 40 49                             cos(t)=± 40 49 =± 40 7 =± 2 10 7 Because the angle is in the second quadrant, we know the x-value is a negative real number, so the cosine is also negative. So<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>cos</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 10 7 If <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 24 25 and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>t</mi></annotation-xml></semantics></math> is in the fourth quadrant, find <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>sin</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 7 25 Finding Sines and Cosines of Special Angles We have already learned some properties of the special angles, such as the conversion from radians to degrees. We can also calculate sines and cosines of the special angles using the Pythagorean Identity and our knowledge of triangles. Finding Sines and Cosines of 45° Angles First, we will look at angles of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>45°</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 4 , as shown in [link] . A<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>45°</mn><mo>–</mo><mn>45°</mn><mo>–</mo><mn>90°</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>triangle is an isosceles triangle, so the x- andy-coordinates of the corresponding point on the circle are the same. Because the x- and y-values are the same, the sine and cosine values will also be equal.

<figure class="small" id="Figure_05_02_009"> </figure>

At<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 4, which is 45 degrees, the radius of the unit circle bisects the first quadrantal angle. This means the radius lies along the line<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>x</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>A unit circle has a radius equal to 1. So, the right triangle formed below the line<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>has sides<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext><mo stretchy="false">(</mo><mi>y</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></annotation-xml></semantics></math> and a radius = 1. See [link].

<figure class="small" id="Figure_05_02_018"> </figure>

From the Pythagorean Theorem we get <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 + y 2 =1 Substituting<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>x</mi><mo>,</mo></mrow></annotation-xml></semantics></math> we get <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 + x 2 =1 Combining like terms we get <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><msup/></mrow></annotation-xml></semantics></math> x 2 =1 And solving for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>,</mo></mrow></annotation-xml></semantics></math> we get <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable><mtr><mtd><mrow><msup><mi>x</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 = 1 2         x=± 1 2 In quadrant I,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 .  At<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 4  or 45 degrees, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 2 , 1 2 ) x= 1 2 ,y= 1 2 cos t= 1 2 ,sin t= 1 2 If we then rationalize the denominators, we get <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>cos</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 2 2 2            = 2 2  sin t= 1 2 2 2             = 2 2 Therefore, the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>coordinates of a point on a circle of radius<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>at an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>45°</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2 2 , 2 2 ).  Finding Sines and Cosines of 30° and 60° Angles Next, we will find the cosine and sine at an angle of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>30°</mn><mo>,</mo></mrow></annotation-xml></semantics></math>or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 6  . First, we will draw a triangle inside a circle with one side at an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>30°</mn><mo>,</mo></mrow></annotation-xml></semantics></math> and another at an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>−30°</mn><mo>,</mo></mrow></annotation-xml></semantics></math> as shown in [link]. If the resulting two right triangles are combined into one large triangle, notice that all three angles of this larger triangle will be<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>60°</mn><mo>,</mo></mrow></annotation-xml></semantics></math> as shown in [link].

<figure class="small" id="Figure_05_02_010"> </figure>

 

<figure class="small" id="Figure_05_02_011"> </figure>

Because all the angles are equal, the sides are also equal. The vertical line has length<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>y</mi><mo>,</mo></mrow></annotation-xml></semantics></math> and since the sides are all equal, we can also conclude that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mn>2</mn><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 r. Since<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 6 )= 1 2 r And since<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math> in our unit circle, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> π 6 )= 1 2 (1)             = 1 2 Using the Pythagorean Identity, we can find the cosine value. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> cos 2 π 6 + sin 2 ( π 6 )=1        cos 2 ( π 6 )+ ( 1 2 ) 2 =1                   cos 2 ( π 6 )= 3 4 Use the square root property.                   cos( π 6 )= ± 3 ± 4 = 3 2 Since y is positive, choose the positive root. The<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>coordinates for the point on a circle of radius<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>at an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>30°</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3 2 , 1 2 ).  At <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 3 (60°), the radius of the unit circle, 1, serves as the hypotenuse of a 30-60-90 degree right triangle,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mi>A</mi><mi>D</mi><mo>,</mo></mrow></annotation-xml></semantics></math>as shown in [link]. Angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>has measure<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>60°</mn><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>At point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>,</mo></mrow></annotation-xml></semantics></math> we draw an angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with measure of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>60°</mn><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>We know the angles in a triangle sum to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>180°</mn><mo>,</mo></mrow></annotation-xml></semantics></math> so the measure of angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is also<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>60°</mn><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Now we have an equilateral triangle. Because each side of the equilateral triangle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is the same length, and we know one side is the radius of the unit circle, all sides must be of length 1.

<figure class="small" id="Figure_05_02_019"> </figure>

The measure of angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is 30°. So, if double, angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is 60°.<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is the perpendicular bisector of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>C</mi><mo>,</mo></mrow></annotation-xml></semantics></math> so it cuts<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in half. This means that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 1 2  the radius, or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 1 2 . Notice that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is the x-coordinate of point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>,</mo></mrow></annotation-xml></semantics></math> which is at the intersection of the 60° angle and the unit circle. This gives us a triangle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mi>A</mi><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with hypotenuse of 1 and side<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>of length<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 1 2 .  From the Pythagorean Theorem, we get <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>x</mi></msup></mrow></annotation-xml></semantics></math> 2 + y 2 =1 Substituting<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 , we get <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></annotation-xml></semantics></math> 1 2 ) 2 + y 2 =1 Solving for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>,</mo></mrow></annotation-xml></semantics></math> we get <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mfrac><mn>1</mn></mfrac></mtd></mtr></mtable></annotation-xml></semantics></math> 4 + y 2 =1         y 2 =1− 1 4         y 2 = 3 4          y=± 3 2 Since<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 3  has the terminal side in quadrant I where the y-coordinate is positive, we choose<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2 ,the positive value. At <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 3  (60°), the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>coordinates for the point on a circle of radius<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>at an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>60°</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1 2 , 3 2 ), so we can find the sine and cosine. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 2 , 3 2 ) x= 1 2 ,y= 3 2 cos t= 1 2 ,sin t= 3 2 We have now found the cosine and sine values for all of the most commonly encountered angles in the first quadrant of the unit circle. [link] summarizes these values. Angle 0 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 6 , or 30 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 4 ,or 45° <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 3 ,or 60° <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 2 , or 90° Cosine 1 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 2   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 2 0 Sine 0 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 1 2   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 2   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 1 [link] shows the common angles in the first quadrant of the unit circle.

<figure id="Figure_05_02_021"> </figure>

Using a Calculator to Find Sine and Cosine To find the cosine and sine of angles other than the special angles, we turn to a computer or calculator. Be aware: Most calculators can be set into “degree” or “radian” mode, which tells the calculator the units for the input value. When we evaluate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mn>30</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>on our calculator, it will evaluate it as the cosine of 30 degrees if the calculator is in degree mode, or the cosine of 30 radians if the calculator is in radian mode. Given an angle in radians, use a graphing calculator to find the cosine. If the calculator has degree mode and radian mode, set it to radian mode. Press the COS key. Enter the radian value of the angle and press the close-parentheses key ")". Press ENTER. Using a Graphing Calculator to Find Sine and Cosine Evaluate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5π 3 ) using a graphing calculator or computer. Enter the following keystrokes: <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>COS</mtext><mo stretchy="false">(</mo><mtext> </mtext><mn>5</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mi>π</mi><mtext> </mtext><mo>÷</mo><mtext> 3 ) ENTER</mtext></mrow></annotation-xml></semantics></math> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5π 3 )=0.5 Analysis We can find the cosine or sine of an angle in degrees directly on a calculator with degree mode. For calculators or software that use only radian mode, we can find the sign of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>20°</mn><mo>,</mo></mrow></annotation-xml></semantics></math> for example, by including the conversion factor to radians as part of the input: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>SIN</mtext><mo stretchy="false">(</mo><mtext> 20 </mtext><mo>×</mo><mtext> </mtext><mi>π</mi><mtext> </mtext><mo>÷</mo><mtext> 180 ) ENTER</mtext></mrow></annotation-xml></semantics></math> Evaluate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 3 ). approximately 0.866025403 Identifying the Domain and Range of Sine and Cosine Functions Now that we can find the sine and cosine of an angle, we need to discuss their domains and ranges. What are the domains of the sine and cosine functions? That is, what are the smallest and largest numbers that can be inputs of the functions? Because angles smaller than 0 and angles larger than<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>can still be graphed on the unit circle and have real values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>, </mo><mi>y</mi><mo>,</mo></mrow></annotation-xml></semantics></math> and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>,</mo></mrow></annotation-xml></semantics></math> there is no lower or upper limit to the angles that can be inputs to the sine and cosine functions. The input to the sine and cosine functions is the rotation from the positive x-axis, and that may be any real number. What are the ranges of the sine and cosine functions? What are the least and greatest possible values for their output? We can see the answers by examining the unit circle, as shown in [link]. The bounds of the x-coordinate are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">[</mo><mn>−1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>The bounds of the y-coordinate are also<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">[</mo><mn>−1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Therefore, the range of both the sine and cosine functions is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">[</mo><mn>−1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>

<figure class="small" id="Figure_05_02_013"> </figure>

Finding Reference Angles We have discussed finding the sine and cosine for angles in the first quadrant, but what if our angle is in another quadrant? For any given angle in the first quadrant, there is an angle in the second quadrant with the same sine value. Because the sine value is the y-coordinate on the unit circle, the other angle with the same sine will share the same y-value, but have the opposite x-value. Therefore, its cosine value will be the opposite of the first angle’s cosine value. Likewise, there will be an angle in the fourth quadrant with the same cosine as the original angle. The angle with the same cosine will share the same x-value but will have the opposite y-value. Therefore, its sine value will be the opposite of the original angle’s sine value. As shown in [link], angle <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>α</mi><mtext>  </mtext></mrow></annotation-xml></semantics></math>has the same sine value as angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>  </mtext><mi>t</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>the cosine values are opposites. Angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>β</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>has the same cosine value as angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>the sine values are opposites. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mtext> </mtext><mi>sin</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> and cos(t)=−cos(α) sin(t)=−sin(β) and cos(t)=    cos(β)

<figure id="Figure_05_02_014"> </figure>

Recall that an angle’s reference angle is the acute angle,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>  </mtext><mi>t</mi><mo>,</mo></mrow></annotation-xml></semantics></math> formed by the terminal side of the angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and the horizontal axis. A reference angle is always an angle between<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>90°</mn><mo>,</mo></mrow></annotation-xml></semantics></math> or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 2  radians. As we can see from [link], for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.

<figure id="Figure_05_02_020"> </figure>

Given an angle between<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mo>,</mo></mrow></annotation-xml></semantics></math> find its reference angle. An angle in the first quadrant is its own reference angle. For an angle in the second or third quadrant, the reference angle is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> π−t | or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> 180°−t |.  For an angle in the fourth quadrant, the reference angle is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mo>−</mo><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>360°</mn><mi>−t</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> If an angle is less than<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>or greater than<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mo>,</mo></mrow></annotation-xml></semantics></math> add or subtract<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>as many times as needed to find an equivalent angle between<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> Finding a Reference Angle Find the reference angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>225°</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>as shown in [link].

<figure class="small" id="Figure_05_02_016"> </figure>

Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>225°</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>is in the third quadrant, the reference angle is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> ( 180°−225° ) |=| −45° |=45° Find the reference angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 5π 3 . <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 3   Using Reference Angles Now let’s take a moment to reconsider the Ferris wheel introduced at the beginning of this section. Suppose a rider snaps a photograph while stopped twenty feet above ground level. The rider then rotates three-quarters of the way around the circle. What is the rider’s new elevation? To answer questions such as this one, we need to evaluate the sine or cosine functions at angles that are greater than 90 degrees or at a negative angle. Reference angles make it possible to evaluate trigonometric functions for angles outside the first quadrant. They can also be used to find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) coordinates for those angles. We will use the reference angle of the angle of rotation combined with the quadrant in which the terminal side of the angle lies. Using Reference Angles to Evaluate Trigonometric Functions We can find the cosine and sine of any angle in any quadrant if we know the cosine or sine of its reference angle. The absolute values of the cosine and sine of an angle are the same as those of the reference angle. The sign depends on the quadrant of the original angle. The cosine will be positive or negative depending on the sign of the x-values in that quadrant. The sine will be positive or negative depending on the sign of the y-values in that quadrant. Using Reference Angles to Find Cosine and Sine Angles have cosines and sines with the same absolute value as their reference angles. The sign (positive or negative) can be determined from the quadrant of the angle. Given an angle in standard position, find the reference angle, and the cosine and sine of the original angle. Measure the angle between the terminal side of the given angle and the horizontal axis. That is the reference angle. Determine the values of the cosine and sine of the reference angle. Give the cosine the same sign as the x-values in the quadrant of the original angle. Give the sine the same sign as the y-values in the quadrant of the original angle. Using Reference Angles to Find Sine and Cosine Using a reference angle, find the exact value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mn>150°</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mo stretchy="false">(</mo><mn>150°</mn><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> Using the reference angle, find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 5π 4  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 5π 4 . 150° is located in the second quadrant. The angle it makes with the x-axis is 180° − 150° = 30°, so the reference angle is 30°. This tells us that 150° has the same sine and cosine values as 30°, except for the sign. We know that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mn>30°</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2      and     sin(30°)= 1 2 .  Since 150° is in the second quadrant, the x-coordinate of the point on the circle is negative, so the cosine value is negative. The y-coordinate is positive, so the sine value is positive. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mn>150°</mn><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2       and     sin(150°)= 1 2   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 5π 4  is in the third quadrant. Its reference angle is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 5π 4 −π= π 4 . The cosine and sine of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 4  are both<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 2 2 . In the third quadrant, both<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are negative, so: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 5π 4 =− 2 2        and      sin  5π 4 =− 2 2 Use the reference angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>315°</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>to find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mn>315°</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo><mn>315°</mn><mo stretchy="false">)</mo></mrow><mo>.</mo></mrow></annotation-xml></semantics></math> Use the reference angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> π 6  to find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> − π 6 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> − π 6 ).  <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>cos</mtext><mo stretchy="false">(</mo><mn>315</mn><mo>°</mo><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 2 , sin(315°)= – 2 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mo stretchy="false">(</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> π 6 )= 3 2 , sin( − π 6 )=− 1 2 Using Reference Angles to Find Coordinates Now that we have learned how to find the cosine and sine values for special angles in the first quadrant, we can use symmetry and reference angles to fill in cosine and sine values for the rest of the special angles on the unit circle. They are shown in [link]. Take time to learn the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>coordinates of all of the major angles in the first quadrant.

<figure id="Figure_05_02_017"> <figcaption>Special angles and coordinates of corresponding points on the unit circle</figcaption> </figure>

In addition to learning the values for special angles, we can use reference angles to find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) coordinates of any point on the unit circle, using what we know of reference angles along with the identities <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>x</mi><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>t</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> y=sin t First we find the reference angle corresponding to the given angle. Then we take the sine and cosine values of the reference angle, and give them the signs corresponding to the y- and x-values of the quadrant. Given the angle of a point on a circle and the radius of the circle, find the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math> coordinates of the point. Find the reference angle by measuring the smallest angle to the x-axis. Find the cosine and sine of the reference angle. Determine the appropriate signs for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> in the given quadrant. Using the Unit Circle to Find Coordinates Find the coordinates of the point on the unit circle at an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 7π 6 . We know that the angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 7π 6  is in the third quadrant. First, let’s find the reference angle by measuring the angle to the x-axis. To find the reference angle of an angle whose terminal side is in quadrant III, we find the difference of the angle and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>π</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>7</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 6 −π= π 6 Next, we will find the cosine and sine of the reference angle: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 6 )= 3 2         sin( π 6 )= 1 2 We must determine the appropriate signs for x and y in the given quadrant. Because our original angle is in the third quadrant, where both<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are negative, both cosine and sine are negative. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 7π 6 )=− 3 2  sin( 7π 6 )=− 1 2 Now we can calculate the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ) coordinates using the identities <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>θ</mi></mrow></annotation-xml></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>.</mo></mrow></annotation-xml></semantics></math> The coordinates of the point are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> − 3 2 ,− 1 2 ) on the unit circle. Find the coordinates of the point on the unit circle at an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 5π 3 .  <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1 2 ,− 3 2 )  Access these online resources for additional instruction and practice with sine and cosine functions. Trigonometric Functions Using the Unit Circle Sine and Cosine from the Unit Circle Sine and Cosine from the Unit Circle and Multiples of Pi Divided by Six Sine and Cosine from the Unit Circle and Multiples of Pi Divided by Four Trigonometric Functions Using Reference Angles Key Equations Cosine <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mi>x</mi></mrow></annotation-xml></semantics></math> Sine <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mi>y</mi></mrow></annotation-xml></semantics></math> Pythagorean Identity <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mi>cos</mi></mrow></msup></mrow></annotation-xml></semantics></math> 2 t+ sin 2 t=1 Key Concepts Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin and has a radius of 1 unit. Using the unit circle, the sine of an angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>equals the y-value of the endpoint on the unit circle of an arc of length<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>whereas the cosine of an angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>  </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>equals the x-value of the endpoint. See [link]. The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an axis. See [link]. When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of special angles. See [link]. Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known. See [link]. The domain of the sine and cosine functions is all real numbers. The range of both the sine and cosine functions is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">[</mo><mi>−</mi><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle. The signs of the sine and cosine are determined from the x- and y-values in the quadrant of the original angle. An angle’s reference angle is the size angle,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>  </mtext><mi>t</mi><mo>,</mo></mrow></annotation-xml></semantics></math> formed by the terminal side of the angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext>  </mtext></mrow></annotation-xml></semantics></math>and the horizontal axis. See[link]. Reference angles can be used to find the sine and cosine of the original angle. See [link]. Reference angles can also be used to find the coordinates of a point on a circle. See [link]. Section Exercises Verbal Describe the unit circle. The unit circle is a circle of radius 1 centered at the origin. What do the x- and y-coordinates of the points on the unit circle represent? Discuss the difference between a coterminal angle and a reference angle. Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>,</mo></mrow></annotation-xml></semantics></math>formed by the terminal side of the angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>  </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and the horizontal axis. Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle. Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle. The sine values are equal. Algebraic For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>t</mi></mrow></annotation-xml></semantics></math> lies. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>sin</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo><</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>cos</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo><</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>sin</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow></annotation-xml></semantics></math> I <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow></annotation-xml></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mrow></annotation-xml></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow></annotation-xml></semantics></math> IV For the following exercises, find the exact value of each trigonometric function. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 6 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>π</mi></mrow></annotation-xml></semantics></math> 0 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 3π 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mi>π</mi></mrow></annotation-xml></semantics></math> −1 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mn>0</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 6 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mn>0</mn></mrow></annotation-xml></semantics></math> Numeric For the following exercises, state the reference angle for the given angle. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>240°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>60°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>170°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>100°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>80°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>315°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>135°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>45°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>2</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 6 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mo>−</mo><mn>11</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mo>−</mo><mtext> </mtext><mn>7</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mo>−</mo><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 8 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 8 For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>225°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>300°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>60°</mn><mo>,</mo></mrow></annotation-xml></semantics></math> Quadrant IV, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>sin</mtext><mo stretchy="false">(</mo><mn>300°</mn><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2 ,cos(300°)= 1 2   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>320°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>135°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>45°</mn><mo>,</mo></mrow></annotation-xml></semantics></math> Quadrant II, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mo stretchy="false">(</mo><mn>135°</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 2 ,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mn>135°</mn><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>210°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>120°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>60°</mn><mo>,</mo></mrow></annotation-xml></semantics></math> Quadrant II, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mo stretchy="false">(</mo><mn>120°</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2 ,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mn>120°</mn><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>250°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>150°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>30°</mn><mo>,</mo></mrow></annotation-xml></semantics></math> Quadrant II, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mo stretchy="false">(</mo><mn>150°</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 ,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mn>150°</mn><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>7</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 6 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 6 , Quadrant III, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mo stretchy="false">(</mo><mfrac/></mrow></annotation-xml></semantics></math> 7π 6 )=− 1 2 , <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>cos</mtext><mo stretchy="false">(</mo><mfrac/></mrow></annotation-xml></semantics></math> 7π 6 )=− 3 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>3</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 4 , Quadrant II, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mo stretchy="false">(</mo><mfrac/></mrow></annotation-xml></semantics></math> 3π 4 )= 2 2 ,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4π 3 )=− 2 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>4</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>2</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 3 , Quadrant II, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mo stretchy="false">(</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π 3 )= 3 2 ,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2π 3 )=− 1 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 6 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>7</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 4 , Quadrant IV, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mo stretchy="false">(</mo><mfrac/></mrow></annotation-xml></semantics></math> 7π 4 )=− 2 2 ,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>cos</mtext><mo stretchy="false">(</mo><mfrac/></mrow></annotation-xml></semantics></math> 7π 4 )= 2 2 For the following exercises, find the requested value. If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>cos</mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t )= 1 7  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is in the 4th quadrant, find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math> If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>cos</mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t )= 2 9  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is in the 1st quadrant, find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mrow><mn>77</mn></mrow></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 9 If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t )= 3 8  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is in the 2nd quadrant, find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>cos</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math> If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t )=− 1 4  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is in the 3rd quadrant, find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>cos</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 15 4 Find the coordinates of the point on a circle with radius 15 corresponding to an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>220°</mn><mo>.</mo></mrow></annotation-xml></semantics></math> Find the coordinates of the point on a circle with radius 20 corresponding to an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>120°</mn><mo>.</mo></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −10,10 3 )  Find the coordinates of the point on a circle with radius 8 corresponding to an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 7π 4 . Find the coordinates of the point on a circle with radius 16 corresponding to an angle of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 5π 9 . <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> –2.778,15.757 )  State the domain of the sine and cosine functions. State the range of the sine and cosine functions. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> –1,1 ]  Graphical For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> .</mtext></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 ,cost=− 3 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 2 ,cos t=− 2 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2 ,cos t=− 1 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 2 ,cos t= 2 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>cos</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mo>−</mo><mn>1</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mo>−</mo><mn>0.596</mn><mo>,</mo><mi>cos</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mn>0.803</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 ,cos t= 3 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 ,cos t= 3 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mn>0.761</mn><mo>,</mo><mi>cos</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mo>−</mo><mn>0.649</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>cos</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mn>0</mn></mrow></annotation-xml></semantics></math> Technology For the following exercises, use a graphing calculator to evaluate. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 5π 9 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 5π 9 −0.1736 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 10 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 10 0.9511 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 3π 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 3π 4 −0.7071 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mn>98°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mn>98°</mn></mrow></annotation-xml></semantics></math> −0.1392 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mtext> </mtext><mn>310°</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mn>310°</mn></mrow></annotation-xml></semantics></math> −0.7660 Extensions <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 11π 3 )cos( −5π 6 ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3π 4 )cos( 5π 3 ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> − 4π 3 )cos( π 2 ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −9π 4 )cos( −π 6 ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 6 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 6 )cos( −π 3 ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7π 4 )cos( −2π 3 ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5π 6 )cos( 2π 3 ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −π 3 )cos( π 4 ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −5π 4 )sin( 11π 6 ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π )sin( π 6 ) 0 Real-World Applications For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,1 ), that is, on the due north position. Assume the carousel revolves counter clockwise. What are the coordinates of the child after 45 seconds? What are the coordinates of the child after 90 seconds? <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,–1 ) What is the coordinates of the child after 125 seconds? When will the child have coordinates<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0.707,–0.707 )  if the ride lasts 6 minutes? (There are multiple answers.) 37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds When will the child have coordinates<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">(</mo><mn>−0.866</mn><mo>,</mo><mn>−0.5</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> if the ride last 6 minutes? Glossary cosine function the x-value of the point on a unit circle corresponding to a given angle Pythagorean Identity a corollary of the Pythagorean Theorem stating that the square of the cosine of a given angle plus the square of the sine of that angle equals 1 sine function the y-value of the point on a unit circle corresponding to a given angle unit circle a circle with a center at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math> and radius 1.