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# 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>[/itex] 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>[/itex] 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>[/itex]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>[/itex]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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex]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>[/itex]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>[/itex]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>[/itex] 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>[/itex]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>[/itex]

<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>[/itex] 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>[/itex]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>[/itex]. 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>[/itex] 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>[/itex] 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>[/itex]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>[/itex] 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>[/itex] 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>[/itex]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>[/itex]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>[/itex] 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>[/itex]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>[/itex]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>[/itex]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>[/itex]

<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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex]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>[/itex] 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>[/itex] 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>[/itex] <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>[/itex] 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>[/itex]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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex]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>[/itex] 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>[/itex]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>[/itex]

<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>[/itex]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>[/itex]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>[/itex] 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>[/itex]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>[/itex] − 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>[/itex] 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>[/itex]

<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>[/itex]

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>[/itex]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>[/itex] 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>[/itex]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>[/itex] 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>[/itex]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>[/itex]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>[/itex] 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>[/itex] <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>[/itex] <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>[/itex] 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>[/itex] 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>[/itex]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>[/itex] 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>[/itex]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>[/itex]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>[/itex] 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>[/itex] 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>[/itex] <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>[/itex] 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>[/itex]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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex]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>[/itex] 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>[/itex]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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] <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>[/itex] 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>[/itex]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>[/itex] 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>[/itex]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>[/itex] 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>[/itex]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>[/itex]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>[/itex]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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex]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>[/itex]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>[/itex]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>[/itex] 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>[/itex]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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex]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>[/itex] 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>[/itex], <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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex]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>[/itex]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>[/itex]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>[/itex] 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>[/itex] 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>[/itex]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>[/itex]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>[/itex]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>[/itex] 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>[/itex]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>[/itex]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>[/itex] 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>[/itex]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>[/itex]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>[/itex]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>[/itex]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>[/itex]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>[/itex]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>[/itex] 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>[/itex]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>[/itex]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>[/itex] 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>[/itex] 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>[/itex]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>[/itex] 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>[/itex]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>[/itex]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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex]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>[/itex]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>[/itex]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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex] 2 1 [link] shows the common angles in the first quadrant of the unit circle.

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

<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>[/itex]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>[/itex]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>[/itex]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>[/itex]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>[/itex] 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>[/itex] 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>[/itex]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>[/itex]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>[/itex] 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>[/itex]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>[/itex] 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>[/itex]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>[/itex] 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>[/itex] 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>[/itex] 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>[/itex]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>[/itex] 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>[/itex]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>[/itex] 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>[/itex]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>[/itex]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>[/itex] 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>[/itex]as shown in [link].

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