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Mathematics LibreTexts

5-3. The Other Trigonometric Functions

The Other Trigonometric Functions
In this section, you will:
  • Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent 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 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> 4 , 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 6 .
  • Use reference angles to evaluate the trigonometric functions secant, cosecant, tangent, and cotangent.
  • Use properties of even and odd trigonometric functions.
  • Recognize and use fundamental identities.
  • Evaluate trigonometric functions with a calculator.

A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent 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 12  or less, regardless of its length. A tangent represents a ratio, so this means that for every 1 inch of rise, the ramp must have 12 inches of run. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions. We have already defined the sine and cosine functions of an angle. Though sine and cosine are the trigonometric functions most often used, there are four others. Together they make up the set of six trigonometric functions. In this section, we will investigate the remaining functions.

Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent

To define the remaining functions, we will once again draw a unit circle with 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 ) 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]. As with the sine and cosine, we can use 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 to find the other functions.

<figure class="small" id="Figure_05_03_001">Graph of circle with angle of t inscribed. Point of (x, y) is at intersection of terminal side of angle and edge of circle.</figure>

The first function we will define is the tangent. The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle. In [link], the tangent of 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>is equal to<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> y x ,x≠0. Because the y-value is equal to the sine 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> and the x-value is equal to 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></mrow></annotation-xml></semantics></math> the tangent of 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>can also be defined as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> cos t ,cos t≠0.The tangent function is abbreviated as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>tan</mtext><mtext>.</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>The remaining three functions can all be expressed as reciprocals of functions we have already defined.

  • The secant function is the reciprocal of the cosine function. In [link], the secant of 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>is equal to<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 cos t = 1 x ,x≠0. The secant function is abbreviated as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sec</mtext><mtext>.</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>
  • The cotangent function is the reciprocal of the tangent function. In [link], the cotangent of 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>is equal to<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> cos t sin t = x y , y≠0. The cotangent function is abbreviated as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>cot</mtext><mtext>.</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>
  • The cosecant function is the reciprocal of the sine function. In [link], the cosecant of 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>is equal to<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 sin t = 1 y ,y≠0. The cosecant function is abbreviated as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>csc</mtext><mtext>.</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>
Tangent, Secant, Cosecant, and Cotangent 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<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>is a point where the terminal side of 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><mtext> </mtext></mrow></annotation-xml></semantics></math>radians intercepts the unit circle, then

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mi>tan</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mtd></mtr></mtable></annotation-xml></semantics></math> y x ,x≠0 sec t= 1 x ,x≠0 csc t= 1 y ,y≠0 cot t= x y ,y≠0
Finding Trigonometric Functions from a Point on the Unit Circle

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> − 3 2 , 1 2 ) is on the unit circle, 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>sin</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>cos</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>tan</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>sec</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>csc</mi><mtext> </mtext><mi>t</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>cot</mi><mtext> </mtext><mi>t</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<figure class="small" id="Figure_05_03_002">Graph of circle with angle of t inscribed. Point of (negative square root of 3 over 2, 1/2) is at intersection of terminal side of angle and edge of circle.</figure>

Because we know 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 the point on the unit circle indicated by 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 use those coordinates to find the six functions:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mi>y</mi><mo>=</mo><mfrac/></mtd></mtr></mtable></annotation-xml></semantics></math> 1 2 cos t=x=− 3 2 tan t= y x = 1 2 − 3 2 = 1 2 ( − 2 3 )=− 1 3 =− 3 3 sec t= 1 x = 1 − 3 2 =− 2 3 =− 2 3 3 csc t= 1 y =1 1 2 =2 cot t= x y = − 3 2 1 2 =− 3 2 ( 2 1 )=− 3

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> 2 2 ,− 2 2 ) is on the unit circle, 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>sin</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>cos</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>tan</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>sec</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>csc</mi><mtext> </mtext><mi>t</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>cot</mi><mtext> </mtext><mi>t</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<figure class="small" id="Figure_05_02_003">Graph of circle with angle of t inscribed. Point of (square root of 2 over 2, negative square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.</figure>

 

<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 ,tan t=−1,sec t= 2 ,csc t=− 2 ,cot t=−1

Finding the Trigonometric Functions of an Angle

Find<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>cos</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>tan</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>sec</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>csc</mi><mtext> </mtext><mi>t</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>cot</mi><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>when<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 6 .

We have previously used the properties of equilateral triangles to demonstrate that<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> π 6 = 1 2  and<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> π 6 = 3 2 . We can use these values and the definitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to find the remaining function values.

 

 

<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>tan</mi><mtext> </mtext><mfrac/></mtd></mtr></mtable></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> π 6 = sin  π 6 cos  π 6              = 1 2 3 2 = 1 3 = 3 3
<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>sec</mi><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> π 6 = 1 cos π 6             = 1 3 2 = 2 3 = 2 3 3
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>csc</mi><mfrac/></annotation-xml></semantics></math> π 6 = 1 sin π 6 = 1 1 2 =2
<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>cot</mi><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> π 6 = cos π 6 sin π 6             = 3 2 1 2 = 3

Find<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>cos</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>tan</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>sec</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>csc</mi><mtext> </mtext><mi>t</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>cot</mi><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>when<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 .

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mi>sin</mi><mfrac/></mtd></mtr></mtable></annotation-xml></semantics></math> π 3 = 3 2 cos π 3 = 1 2 tan π 3 = 3 sec π 3 =2 csc π 3 = 2 3 3 cot π 3 = 3 3

Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting<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>equal to the cosine 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>equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in [link].

Angle <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> <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 , or 30° <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 , or 45° <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 , or 60° <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> 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><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><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
Tangent 0 <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> 3 1 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msqrt><mn>3</mn></msqrt></mrow></annotation-xml></semantics></math> Undefined
Secant 1 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>2</mn><msqrt/></mrow></mfrac></mrow></annotation-xml></semantics></math> 3 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msqrt><mn>2</mn></msqrt></mrow></annotation-xml></semantics></math> 2 Undefined
Cosecant Undefined 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msqrt><mn>2</mn></msqrt></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>2</mn><msqrt/></mrow></mfrac></mrow></annotation-xml></semantics></math> 3 3 1
Cotangent Undefined <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msqrt><mn>3</mn></msqrt></mrow></annotation-xml></semantics></math> 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> 3 0

Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent

We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x- and y-values in the original quadrant.[link] shows which functions are positive in which quadrant.

To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase “A Smart Trig Class.” Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is “A,” all of the six trigonometric functions are positive. In quadrant II, “Smart,” only sine and its reciprocal function, cosecant, are positive. In quadrant III, “Trig,” only tangent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, “Class,” only cosine and its reciprocal function, secant, are positive.

<figure class="small" id="Figure_05_03_004">Graph of circle with each quadrant labeled. Under quadrant 1, labels fro sin t, cos t, tan t, sec t, csc t, and cot t. Under quadrant 2, labels for sin t and csc t. Under quadrant 3, labels for tan t and cot t. Under quadrant 4, labels for cos t, sec t.</figure>

Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions.

  1. Measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference angle.
  2. Evaluate the function at the reference angle.
  3. Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative.
Using Reference Angles to Find Trigonometric Functions

Use reference angles to find all six trigonometric functions 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> 5π 6 . 

The angle between this angle’s terminal side and the x-axis 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> π 6 , so that is the reference angle. Since<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> 5π 6  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, cosine, sine, secant, and cosecant will be negative, while tangent and cotangent will be positive.

<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> − 5π 6 )=− 3 2 ,sin( − 5π 6 )=− 1 2 ,tan( − 5π 6 )= 3 3 sec( − 5π 6 )=− 2 3 3 ,csc( − 5π 6 )=−2,cot( − 5π 6 )= 3

Use reference angles to find all six trigonometric functions 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> 7π 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> −7π 4 )= 2 2 ,cos( −7π 4 )= 2 2 ,tan( −7π 4 )=1,

 
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −7π 4 )= 2 ,csc( −7π 4 )= 2 ,cot( −7π 4 )=1

 

Using Even and Odd Trigonometric Functions

To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.

Consider the function<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>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x 2 , shown in [link]. The graph of the function is symmetrical about the y-axis. All along the curve, any two points with opposite x-values have the same function value. This matches the result of calculation:<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> (4) 2 = (−4) 2 ,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mn>−5</mn><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 2 = (5) 2 , and so on. So<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>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x 2  is an even function, a function such that two inputs that are opposites have the same output. That means<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −x )=f( x ). 

<figure class="medium" id="Figure_05_03_005"> <figcaption>The function<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>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x 2   is an even function.</figcaption> Graph of parabola with points (-2, 4) and (2, 4) labeled.</figure>

Now consider the function<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>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x 3 , shown in [link]. The graph is not symmetrical about the y-axis. All along the graph, any two points with opposite x-values also have opposite y-values. So<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>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x 3  is an odd function, one such that two inputs that are opposites have outputs that are also opposites. That means<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −x )=−f( x ). 

<figure class="small" id="Figure_05_03_006"> <figcaption>The function<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>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x 3   is an odd function.</figcaption> Graph of function with labels for points (-1, -1) and (1, 1).</figure>

We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in [link]. The sine of the positive angle is<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>The sine of the negative angle is −y. The sine function, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in [link].

<figure class="small" id="Figure_05_03_007">Graph of circle with angle of t and -t inscribed. Point of (x, y) is at intersection of terminal side of angle t and edge of circle. Point of (x, -y) is at intersection of terminal side of angle -t and edge of circle.</figure>
<math display="inline" 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><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mi>y</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> sin(−t)=−y         sin t≠sin(−t) <math display="inline" 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><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>cos</mtext><mtext> </mtext><mi>t</mi><mo>=</mo><mi>x</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> cos(−t)=x          cos t=cos(−t) <math display="inline" 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><mtext> </mtext><mtext>tan</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> y x  tan(−t)=− y x          tan t≠tan(−t)
<math display="inline" 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><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>sec</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 x sec(−t)= 1 x         sec t=sec(−t) <math display="inline" 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><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>csc</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 y  csc(−t)= 1 −y          csc t≠csc(−t) <math display="inline" 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><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>cot</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x y  cot(−t)= x −y          cot t≠cot(−t)
Even and Odd Trigonometric Functions

An even function is one in which<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>−</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>

An odd function is one in which<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>−</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>−</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>

Cosine and secant are even:

<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><mo stretchy="false">(</mo><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>cos</mtext><mtext> </mtext><mi>t</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> sec(−t)=sec t

Sine, tangent, cosecant, and cotangent are odd:

<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><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>sin</mi><mtext> </mtext><mi>t</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> tan(−t)=−tan t csc(−t)=−csc t cot(−t)=−cot t
Using Even and Odd Properties of Trigonometric Functions

If the secant of 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>is 2, what is the secant of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>−</mi><mi>t</mi><mo>?</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>

Secant is an even function. The secant of an angle is the same as the secant of its opposite. So if the secant of angle t is 2, the secant of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>−</mi><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is also 2.

If the cotangent of 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>is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msqrt/></mrow></annotation-xml></semantics></math> 3 , what is the cotangent of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>−</mi><mi>t</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><mo>−</mo><msqrt/></mrow></annotation-xml></semantics></math> 3  

Recognizing and Using Fundamental Identities

We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine.

Fundamental Identities

We can derive some useful identities from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>tan</mtext><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac><mrow/></mfrac></mrow></annotation-xml></semantics></math> sin tcos  t
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 cos t
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 sin t
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>cot</mtext><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 tan t = cos t sin t
Using Identities to Evaluate Trigonometric Functions
  1. Given <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><mn>45°</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 2 ,cos(45°)= 2 2 , evaluate <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mo stretchy="false">(</mo><mn>45°</mn><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>
  2. Given<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> 5π 6 )= 1 2 ,cos( 5π 6 )=− 3 2 ,evaluate sec( 5π 6 ).

Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions.

  1. <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>tan</mi><mo stretchy="false">(</mo><mn>45°</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> sin(45°) cos(45°)                    = 2 2 2 2                    =1
  2. <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>sec</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 5π 6 )= 1 cos( 5π 6 )                    = 1 − 3 2                    = −2 3 1                    = −2 3                     =− 2 3 3

Evaluate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>csc</mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7π 6 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>2</mn></mrow></annotation-xml></semantics></math>

Using Identities to Simplify Trigonometric Expressions

Simplify <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>sec</mi><mtext> </mtext><mi>t</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> tan t .

We can simplify this by rewriting both functions in terms of 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><mfrac><mrow><mi>sec</mi><mtext> </mtext><mi>t</mi></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> tan t = 1 cos t sin t cos t To divide the functions, we multiply by the reciprocal.             = 1 cos t cos t sin tDivide out the cosines.             = 1 sin t Simplify and use the identity.             =csc t

By showing that<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> sec t tan t  can be simplified to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>csc</mi><mtext> </mtext><mi>t</mi><mo>,</mo></mrow></annotation-xml></semantics></math>we have, in fact, established a new identity.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>sec</mi><mtext> </mtext><mi>t</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> tan t =csc t

Simplify<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>t</mi><mo stretchy="false">(</mo><mi>cos</mi><mtext> </mtext><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><mtext> </mtext><mi>t</mi></mrow></annotation-xml></semantics></math>

Alternate Forms of the Pythagorean Identity

We can use these fundamental identities to derive alternative forms of the Pythagorean Identity,<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. One form is obtained by dividing both sides by<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:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable><mtr><mtd><mrow><mfrac><mrow><msup><mrow><mi>cos</mi></mrow></msup></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 t cos 2 t + sin 2 t cos 2 t = 1 cos 2 t              1+ tan 2 t= sec 2 t

The other form is obtained by dividing both sides by <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> sin 2 t:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable><mtr><mtd><mrow><mfrac><mrow><msup><mrow><mi>cos</mi></mrow></msup></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 t sin 2 t + sin 2 t sin 2 t = 1 sin 2 t              cot 2 t+1= csc 2 t
Alternate Forms of the Pythagorean Identity
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>+</mo><msup/></mrow></annotation-xml></semantics></math> tan 2 t= sec 2 t
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mi>cot</mi></mrow></msup></mrow></annotation-xml></semantics></math> 2 t+1= csc 2 t
Using Identities to Relate Trigonometric Functions

If<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><mfrac/></mrow></annotation-xml></semantics></math> 12 13  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 quadrant IV, as shown in [link], find the values of the other five trigonometric functions.

<figure class="small" id="Figure_05_03_008">Graph of circle with angle of t inscribed. Point of (12/13, y) is at intersection of terminal side of angle and edge of circle.</figure>

We can find the sine using the Pythagorean Identity,<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, and the remaining functions by relating them to 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><msup><mrow><mrow><mo>(</mo></mrow></mrow></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 12 13 ) 2 + sin 2 t=1               sin 2 t=1− ( 12 13 ) 2               sin 2 t=1− 144 169               sin 2 t= 25 169                sin t=± 25169                sin t=± 25 169                sin t=± 5 13

The sign of the sine depends on the y-values in the quadrant where the angle is located. Since the angle is in quadrant IV, where the y-values are negative, its sine is negative,<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> 5 13 .

The remaining functions can be calculated using identities relating them to sine and cosine.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mtd></mtr></mtable></annotation-xml></semantics></math> sin t cos t = − 5 13 12 13 =− 5 12 sec t= 1 cos t = 1 12 13 = 13 12 csc t= 1 sin t = 1 − 5 13 =− 13 5 cot t= 1 tan t = 1 − 512 =− 12 5

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sec</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 17 8  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>0</mn><mo><</mo><mi>t</mi><mo><</mo><mi>π</mi><mo>,</mo></mrow></annotation-xml></semantics></math> find the values of the other five functions.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mspace width="0.2em"/><mi>t</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 8 17 ,sint= 15 17 ,tant=− 15 8

 
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mspace width="0.2em"/><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 17 15 ,cott=− 8 15

 

As we discussed in the chapter opening, a function that repeats its values in regular intervals is known as a periodic function. The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or<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> will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.

Other functions can also be periodic. For example, the lengths of months repeat every four years. If<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>represents the length time, measured in years, 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>x</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>represents the number of days in February, then<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>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period. A periodis the shortest interval over which a function completes one full cycle—in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.

Period of a Function

The period<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>of a repeating function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is the number representing the interval such that<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>x</mi><mo>+</mo><mi>P</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>for any value 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><mtext> </mtext></mrow></annotation-xml></semantics></math>

The period of the cosine, sine, secant, and cosecant functions 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><mtext> </mtext></mrow></annotation-xml></semantics></math>

The period of the tangent and cotangent functions is<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></mrow></annotation-xml></semantics></math>

Finding the Values of Trigonometric Functions

Find the values of the six trigonometric functions of 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>based on [link].

<figure class="small" id="Figure_05_03_009">Graph of circle with angle of t inscribed. Point of (1/2, negative square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.</figure>
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mi>y</mi><mo>=</mo><mo>−</mo><mfrac/></mtd></mtr></mtable></annotation-xml></semantics></math> 3 2 cos t=x=− 1 2 tan t= sint cost = − 3 2 − 1 2 = 3 sec t= 1 cost = 1 − 1 2 =−2 csc t= 1 sint = 1 − 3 2 =− 2 3 3 cot t=1 tant = 1 3 = 3 3

Find the values of the six trigonometric functions of 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> based on [link].

<figure class="small" id="Figure_05_03_010">Graph of circle with angle of t inscribed. Point of (0, -1) is at intersection of terminal side of angle and edge of circle.</figure>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mo>−</mo><mn>1</mn><mo>,</mo><mi>cos</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>tan</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mtext>Undefined</mtext></mtd></mtr></mtable></annotation-xml></semantics></math> sec t= Undefined,csc t=−1,cot t=0

Finding the Value of Trigonometric Functions

If<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 )=− 3 2  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><mfrac/></mrow></annotation-xml></semantics></math> 1 2 ,find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sec</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mtext>csc</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mtext>tan</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> cot</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><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><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mtable columnalign="left"><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mi>sec</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mtd></mtr></mtable></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 cos t = 1 1 2 =2 csc t= 1 sin t = 1 − 3 2 − 2 3 3 tan t= sin t cos t = − 3 2 1 2 =− 3 cot t= 1 tan t = 1 − 3 =− 3 3

If<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 )= 2 2   and<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 )= 2 2 ,find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sec</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mtext>csc</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mtext>tan</mtext><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> and cot</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>sec</mi><mtext> </mtext><mi>t</mi><mo>=</mo><msqrt/></mrow></annotation-xml></semantics></math> 2 ,csc t= 2 ,tan t=1,cot t=1

Evaluating Trigonometric Functions with a Calculator

We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.

Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.

If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor<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 180  to convert the degrees to radians. To find the secant 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> we could press

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>(for a scientific calculator):</mtext><mfrac/></mrow></annotation-xml></semantics></math> 1 30× π 180 COS

or

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>(for a graphing calculator):</mtext><mtext> </mtext><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 1 cos( 30π 180 )

Given an angle measure in radians, use a scientific calculator to find the cosecant.

  1. If the calculator has degree mode and radian mode, set it to radian mode.
  2. Enter:<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><mtext> </mtext></mrow></annotation-xml></semantics></math>
  3. Enter the value of the angle inside parentheses.
  4. Press the SIN key.
  5. Press the = key.

Given an angle measure in radians, use a graphing utility/calculator to find the cosecant.

  1. If the graphing utility has degree mode and radian mode, set it to radian mode.
  2. Enter:<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><mtext> </mtext></mrow></annotation-xml></semantics></math>
  3. Press the SIN key.
  4. Enter the value of the angle inside parentheses.
  5. Press the ENTER key.
Evaluating the Secant Using Technology

Evaluate the cosecant 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 7 . 

For a scientific calculator, enter information as follows:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>1 / ( 5 </mtext><mo>×</mo><mtext> </mtext><mi>π</mi><mtext> / 7 ) SIN =</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>csc</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5π 7 )≈1.279

Evaluate the cotangent 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 8 . 

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>≈</mo><mo>−</mo><mn>2.414</mn></mrow></annotation-xml></semantics></math>

Access these online resources for additional instruction and practice with other trigonometric functions.

Key Equations

Tangent function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> sint cost
Secant function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 cost
Cosecant function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 sint
Cotangent function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 tan t = cos t sin t

Key Concepts

  • The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle.
  • The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function.
  • The six trigonometric functions can be found from a point on the unit circle. See [link].
  • Trigonometric functions can also be found from an angle. See [link].
  • Trigonometric functions of angles outside the first quadrant can be determined using reference angles. See [link].
  • A function is said to be even if<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><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and odd if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −x )=−f( x ).
  • Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
  • Even and odd properties can be used to evaluate trigonometric functions. See [link].
  • The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.
  • Identities can be used to evaluate trigonometric functions. See [link] and [link].
  • Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. See [link].
  • The trigonometric functions repeat at regular intervals.
  • The period<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>of a repeating function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is the smallest interval such that<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>x</mi><mo>+</mo><mi>P</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>for any value 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></mrow></annotation-xml></semantics></math>
  • The values of trigonometric functions of special angles can be found by mathematical analysis.
  • To evaluate trigonometric functions of other angles, we can use a calculator or computer software. See [link].

Section Exercises

Verbal

On an interval of<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> 0,2π ), can the sine and cosine values of a radian measure ever be equal? If so, where?

Yes, when the 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> π 4  and the terminal side of the angle is in quadrants I and III. Thus, at<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> π 4 , 5π 4 , the sine and cosine values are equal.

What would you estimate the cosine of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>π</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>degrees to be? Explain your reasoning.

For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?

Substitute the sine of the angle in for<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 Pythagorean Theorem<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. Solve 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 take the negative solution.

Describe the secant function.

Tangent and cotangent have a period of<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> What does this tell us about the output of these functions?

The outputs of tangent and cotangent will repeat every<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>π</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> units.

Algebraic

For the following exercises, find the exact value of each expression.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</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>sec</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><mn>2</mn><msqrt/></mrow></mfrac></mrow></annotation-xml></semantics></math> 3 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</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>cot</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><msqrt><mn>3</mn></msqrt></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</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>sec</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><msqrt><mn>2</mn></msqrt></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</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>cot</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 4

1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</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><mi>sec</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 3

2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</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><mi>cot</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> 3

For the following exercises, use reference angles to evaluate the expression.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 5π 6

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 7π 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> 2 3 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 11π 6

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 13π 6

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msqrt><mn>3</mn></msqrt></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 7π 4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</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><mo>−</mo><msqrt/></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>csc</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 5π 4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 11π 4

−1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 8π 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 4π 3

−2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 2π 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><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><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>tan</mi><mtext> </mtext><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><mi>sec</mi><mtext> </mtext><mn>300°</mn></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>csc</mi><mtext> </mtext><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><mi>cot</mi><mtext> </mtext><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><mfrac><mrow><msqrt><mn>3</mn></msqrt></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><mi>tan</mi><mtext> </mtext><mn>330°</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>sec</mi><mtext> </mtext><mn>120°</mn></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>csc</mi><mtext> </mtext><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><mi>cot</mi><mtext> </mtext><mn>315°</mn></mrow></annotation-xml></semantics></math>

−1

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sin</mtext><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 4 , and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is in quadrant II, 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><mi>t</mi><mo>,</mo><mi>sec</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>csc</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>tan</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>cot</mi><mtext> </mtext><mi>t</mi><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><mtext> </mtext><mi>t</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 3 , and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is in quadrant III, find<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>sec</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>csc</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>tan</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>cot</mi><mtext> </mtext><mi>t</mi><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><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 2 3 ,sec t=−3,csc t=− 3 2 4 ,tan t=2 2 ,cot t= 2 4

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 12 5 , and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>0</mn><mo>≤</mo><mi>t</mi><mo><</mo><mfrac/></mrow></annotation-xml></semantics></math> π 2 , find<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>cos</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>sec</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>csc</mi><mtext> </mtext><mi>t</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>cot</mi><mtext> </mtext><mi>t</mi><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><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2  and<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><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 , find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sec</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>csc</mi><mtext> </mtext><mi>t</mi><mo>,</mo><mi>tan</mi><mtext> </mtext><mi>t</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>cot</mi><mtext> </mtext><mi>t</mi><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>sec</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mn>2</mn><mo>,</mo><mi>csc</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 3 3 , tan t= 3 , cot t= 3 3

If<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><mn>40°</mn><mo>≈</mo><mn>0.643</mn><mtext> </mtext><mtext> </mtext><mi>cos</mi><mtext> </mtext><mn>40°</mn><mo>≈</mo><mn>0.766</mn><mtext> </mtext><mtext> </mtext><mtext>sec</mtext><mtext> </mtext><mn>40°</mn><mo>,</mo><mtext>csc</mtext><mtext> </mtext><mn>40°</mn><mo>,</mo><mtext>tan</mtext><mtext> </mtext><mn>40°</mn><mo>,</mo><mtext>and</mtext><mtext> </mtext><mtext>cot</mtext><mtext> </mtext><mn>40°</mn><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><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 2 , what is the<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>−</mi><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><mtext> </mtext><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 2  

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>cos</mtext><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 , what is the<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>−</mi><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>sec</mtext><mtext> </mtext><mi>t</mi><mo>=</mo><mn>3.1</mn><mo>,</mo></mrow></annotation-xml></semantics></math> what is the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>sec</mtext><mo stretchy="false">(</mo><mi>−</mi><mi>t</mi><mo stretchy="false">)</mo><mo>?</mo></mrow></annotation-xml></semantics></math>

3.1

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>csc</mtext><mtext> </mtext><mi>t</mi><mo>=</mo><mn>0.34</mn><mo>,</mo></mrow></annotation-xml></semantics></math> what is the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>csc</mtext><mo stretchy="false">(</mo><mi>−</mi><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>tan</mtext><mtext> </mtext><mi>t</mi><mo>=</mo><mo>−</mo><mn>1.4</mn><mo>,</mo></mrow></annotation-xml></semantics></math> what is the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>tan</mtext><mo stretchy="false">(</mo><mi>−</mi><mi>t</mi><mo stretchy="false">)</mo><mo>?</mo></mrow></annotation-xml></semantics></math>

1.4

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>cot</mtext><mtext> </mtext><mi>t</mi><mo>=</mo><mn>9.23</mn><mo>,</mo></mrow></annotation-xml></semantics></math> what is the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>cot</mtext><mo stretchy="false">(</mo><mi>−</mi><mi>t</mi><mo stretchy="false">)</mo><mo>?</mo></mrow></annotation-xml></semantics></math>

Graphical

For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.

Graph of circle with angle of t inscribed. Point of (square root of 2 over 2, square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

<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> 2 2 ,cos t= 2 2 ,tan t=1,cot t=1,sec t= 2 ,csc t= 2

Graph of circle with angle of t inscribed. Point of (square root of 3 over 2, 1/2) is at intersection of terminal side of angle and edge of circle.
Graph of circle with angle of t inscribed. Point of (-1/2, negative square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.

<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> 3 2 ,cos t=− 1 2 ,tan t= 3 ,cot t= 3 3 ,sec t=−2,csc t=− 2 3 3

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>csc</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>cot</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 4π 7

–0.228

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</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>tan</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 5π 8

–2.414

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sec</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>csc</mi><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 4

1.414

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>tan</mtext><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>cot</mi><mtext> </mtext><mn>33°</mn></mrow></annotation-xml></semantics></math>

1.540

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cot</mi><mtext> </mtext><mn>140°</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>sec</mi><mtext> </mtext><mn>310°</mn></mrow></annotation-xml></semantics></math>

1.556

Extensions

For the following exercises, use identities to evaluate the expression.

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t )≈2.7, 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> t )≈0.94, 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> t ).

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t )≈1.3, and<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 )≈0.61, find<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 ). 

<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> t )≈0.79

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>csc</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t )≈3.2, and<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 )≈0.95, find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t ).

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t )≈0.58, and<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 )≈0.5, find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>csc</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mi>t</mi><mo>≈</mo><mn>1.16</mn></mrow></annotation-xml></semantics></math>

Determine whether the function<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>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is even, odd, or neither.

Determine whether the function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><msup/></mrow></annotation-xml></semantics></math> sin 2 x cos x + sec x is even, odd, or neither.

even

Determine whether the function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext><mo>−</mo><mn>2</mn><msup/></mrow></annotation-xml></semantics></math> cos 2 x is even, odd, or neither.

Determine whether the function<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>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> csc 2 x+sec x is even, odd, or neither.

even

For the following exercises, use identities to simplify the expression.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>csc</mi><mtext> </mtext><mi>t</mi><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>t</mi></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><mi>sec</mi><mtext> </mtext><mi>t</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> csc t

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>sin</mi><mtext> </mtext><mi>t</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> cos t =tan t

Real-World Applications

The amount of sunlight in a certain city can be modeled by the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mo>=</mo><mn>15</mn><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1 600 d ), where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>represents the hours of sunlight, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>d</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is the day of the year. Use the equation to find how many hours of sunlight there are on February 10, the 42ndday of the year. State the period of the function.

The amount of sunlight in a certain city can be modeled by the function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mo>=</mo><mn>16</mn><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1 500 d ), where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> represents the hours of sunlight, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>d</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the 267th day of the year. State the period of the function.

13.77 hours, period:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>1000</mn><mi>π</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>

The equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>=</mo><mn>20</mn><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2πt )+100 models the blood pressure,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>,</mo></mrow></annotation-xml></semantics></math> 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>represents time in seconds. (a) Find the blood pressure after 15 seconds. (b) What are the maximum and minimum blood pressures?

The height of a piston,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mo>,</mo></mrow></annotation-xml></semantics></math> in inches, can be modeled by the equation<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><mn>2</mn><mi>cos</mi><mtext> </mtext><mi>x</mi><mo>+</mo><mn>6</mn><mo>,</mo></mrow></annotation-xml></semantics></math> where<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>represents the crank angle. Find the height of the piston when the crank angle is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>55°</mn><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>

7.73 inches

The height of a piston,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mo>,</mo></mrow></annotation-xml></semantics></math>in inches, can be modeled by the equation<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><mn>2</mn><mi>cos</mi><mtext> </mtext><mi>x</mi><mo>+</mo><mn>5</mn><mo>,</mo></mrow></annotation-xml></semantics></math>where<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>represents the crank angle. Find the height of the piston when the crank angle is <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>55°</mn><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>

Glossary

cosecant
the reciprocal of the sine function: on the unit circle,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>csc</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 y ,y≠0
cotangent
the reciprocal of the tangent function: on the unit circle,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cot</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x y ,y≠0
identities
statements that are true for all values of the input on which they are defined
period
the smallest interval<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> of a repeating function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>such that<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>x</mi><mo>+</mo><mi>P</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>
secant
the reciprocal of the cosine function: on the unit circle,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sec</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 x ,x≠0
tangent
the quotient of the sine and cosine: on the unit circle,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>t</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> y x ,x≠0