Introduction to Reciprocal Functions

We have discussed three trigonometric functions: sine, cosine, and tangent. Each of these functions has a reciprocal function, which is defined by the reciprocal of the ratio for the original trigonometric function. Note that reciprocal functions differ from inverse functions. Inverse functions are a way of working backwards, or determining an angle given a trigonometric ratio; they involve working with the same ratios as the original function. 

The three reciprocal functions are described below.

Secant

The secant function is the reciprocal of the cosine function, and is abbreviated as $\sec$. It can be described as the ratio of the length of the hypotenuse to the length of the adjacent side in a triangle. 

$\displaystyle{ \begin{aligned} \sec x &= \frac{1}{\cos x} \\ \sec x &= \frac{\text{hypotenuse}}{\text{adjacent}} \end{aligned} }$

It is easy to calculate secant with values in the unit circle. Recall that for any point on the circle, the $x$-value gives $\cos t$ for the associated angle $t$. Therefore, the secant function for that angle is 

$\displaystyle{\sec t = \frac{1}{x}}$

Cosecant

The cosecant function is the reciprocal of the sine function, and is abbreviated as$\csc$. It can be described as the ratio of the length of the hypotenuse to the length of the opposite side in a triangle. 

$\displaystyle{ \begin{aligned} \csc x &= \frac{1}{\sin x} \\ \csc x &= \frac{\text{hypotenuse}}{\text{opposite}} \end{aligned} }$

As with secant, cosecant can be calculated with values in the unit circle. Recall that for any point on the circle, the $y$-value gives $\sin t$. Therefore, the cosecant function for the same angle is 

$\displaystyle{\csc t = \frac{1}{y}}$

Cotangent

The cotangent function is the reciprocal of the tangent function, and is abbreviated as $\cot$. It can be described as the ratio of the length of the adjacent side to the length of the hypotenuse in a triangle. 

$\displaystyle{ \begin{aligned} \cot x &= \frac{1}{\tan x} \\ \cot x &= \frac{\text{adjacent}}{\text{opposite}} \end{aligned} }$

Also note that because $\displaystyle{\tan x = \frac{\sin x}{\cos x}}$, its reciprocal is 

$\displaystyle{\cot x = \frac{\cos x}{\sin x}}$

Cotangent can also be calculated with values in the unit circle. Applying the $x$- and $y$-coordinates associated with angle $t$, we have 

$\displaystyle{ \begin{aligned} \cot t &= \frac{\cos t}{\sin t} \\ \cot t &= \frac{x}{y} \end{aligned} }$

Calculating Reciprocal Functions

We now recognize six trigonometric functions that can be calculated using values in the unit circle. Recall that we used values for the sine and cosine functions to calculate the tangent function. We will follow a similar process for the reciprocal functions, referencing the values in the unit circle for our calculations.

For example, let's find the value of $\sec{\left(\frac{\pi}{3}\right)}$. 

Applying $\displaystyle{\sec x = \frac{1}{\cos x}}$, we can rewrite this as:

$\displaystyle{ \sec{\left(\frac{\pi}{3}\right)}= \frac{1}{\cos{\left({\frac{\pi}{3}}\right)}} }$

From the unit circle, we know that $\displaystyle{\cos{\left({\frac{\pi}{3}}\right)}= \frac{1}{2}}$. Using this, the value of $\displaystyle{ \sec{\left(\frac{\pi}{3}\right)}}$ can be found:

$\displaystyle{ \begin{aligned} \sec{\left(\frac{\pi}{3}\right)} &= \frac{1}{\frac{1}{2}} \\ &= 2 \end{aligned} }$

The other reciprocal functions can be solved in a similar manner. 

Example

Use the unit circle to calculate $\sec t$, $\cot t$, and $\csc t$ at the point $\displaystyle{\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)}$. 

Point on a unit circle

The point $\displaystyle{\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)}$, shown on a unit 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 three functions. 

Recall that the $x$-coordinate gives the value for the cosine function, and the $y$-coordinate gives the value for the sine function. In other words:

$\displaystyle{ \begin{aligned} x &= \cos t \\ &= -\frac{\sqrt{3}}{2} \end{aligned} }$

and

$\displaystyle{ \begin{aligned} y &= \sin t \\ &= \frac{1}{2} \end{aligned} }$

Using this information, the values for the reciprocal functions at angle $t$ can be calculated:

$\displaystyle{ \begin{aligned} \sec t &= \frac{1}{\cos t} \\ &= \frac{1}{x} \\ &= \left(\frac{1}{-\frac{\sqrt{3}}{2}} \right)\\ &= -\frac{2}{\sqrt{3}} \\ &= \left(-\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \right)\\ &= -\frac{2\sqrt{3}}{3} \end{aligned} }$

$\displaystyle{ \begin{aligned} \cot t &= \frac{\cos t}{\sin t} \\ &= \frac{x}{y} \\ &= \left(\frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}}\right) \\ &= \left(-\frac{\sqrt{3}}{2}\cdot \frac{2}{1} \right) \\ &= -\sqrt{3} \end{aligned} }$

$\displaystyle{ \begin{aligned} \csc t &= \frac{1}{\sin t} \\ & = \frac{1}{y} \\ & = \left(\frac{1}{\frac{1}{2}}\right) \\ & = 2 \end{aligned} }$