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In this section you will:
  • Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of π 3 , π 4 , and π 6 .
  • Use reference angles to evaluate the trigonometric functions secant, 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 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

We can also define the remaining functions in terms of the unit circle with a point ( x , y ) corresponding to an angle of t , as shown in [link] . As with the sine and cosine, we can use the ( x , y ) coordinates to find the other functions.

This image is a graph of circle with angle of t inscribed and a radius of 1. Point of (x, y) is at intersection of terminal side of angle and edge of circle.

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 t is equal to y x , x 0. Because the y -value is equal to the sine of t , and the x -value is equal to the cosine of t , the tangent of angle t can also be defined as sin t cos t , cos t 0. The tangent function is abbreviated as tan . 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 t is equal to 1 cos t = 1 x , x 0. The secant function is abbreviated as sec .
  • The cotangent    function is the reciprocal of the tangent function. In [link] , the cotangent of angle t is equal to cos t sin t = x y , y 0. The cotangent function is abbreviated as cot .
  • The cosecant    function is the reciprocal of the sine function. In [link] , the cosecant of angle t is equal to 1 sin t = 1 y , y 0. The cosecant function is abbreviated as csc .

Tangent, secant, cosecant, and cotangent functions

If t is a real number and ( x , y ) is a point where the terminal side of an angle of t radians intercepts the unit circle, then

tan  t = 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 ( 3 2 , 1 2 ) is on the unit circle, as shown in [link] . Find sin t , cos t , tan t , sec t , csc t , and cot t .

This is an image of a graph of circle with angle of t inscribed and with radius 1. Point of (negative square root of 3 over 2, 1/2) is at intersection of terminal side of angle and edge of circle.

Because we know the ( x , y ) coordinates of the point on the unit circle indicated by angle t , we can use those coordinates to find the six functions:

sin  t = y = 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
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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