5.3 The other trigonometric functions

A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is $\frac{1}{12}$ 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 $\left(x,y\right)$ corresponding to an angle of $t,$ as shown in [link] . As with the sine and cosine, we can use the $\left(x,y\right)$ coordinates to find the other functions.

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 $\frac{y}{x},x\mathrm{\ne 0.}$ 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 $\frac{\sin t}{\cos t},\cos t\ne 0.$ The tangent function is abbreviated as $\text{tan}\text{.}$ 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 $\frac{1}{\cos t}=\frac{1}{x},x\ne 0.$ The secant function is abbreviated as $\text{sec}\text{.}$
• The cotangent    function is the reciprocal of the tangent function. In [link] , the cotangent of angle $t$ is equal to $\frac{\cos t}{\sin t}=\frac{x}{y},y\ne 0.$ The cotangent function is abbreviated as $\text{cot}\text{.}$
• The cosecant    function is the reciprocal of the sine function. In [link] , the cosecant of angle $t$ is equal to $\frac{1}{\sin t}=\frac{1}{y},y\ne 0.$ The cosecant function is abbreviated as $\text{csc}\text{.}$

Finding trigonometric functions from a point on the unit circle

The point $\left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right)$ is on the unit circle, as shown in [link] . Find $\sin t,\cos t,\tan t,\sec t,\csc t,$ and $\cot t.$

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:

$$\begin{array}{l}\sin t=y=\frac{1}{2}\\ \cos t=x=-\frac{\sqrt{3}}{2}\\ \tan t=\frac{y}{x}=\frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=\frac{1}{2}\left(-\frac{2}{\sqrt{3}}\right)=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}\\ \sec t=\frac{1}{x}=\frac{1}{\frac{-\frac{\sqrt{3}}{2}}{}}=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}\\ \csc t=\frac{1}{y}=\frac{1}{\frac{1}{2}}=2\\ \cot t=\frac{x}{y}=\frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=-\frac{\sqrt{3}}{2}\left(\frac{2}{1}\right)=-\sqrt{3}\end{array}$$

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