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We have already learned about the reciprocal trigonometric functions (secant, cosecant, and cotangent) and their relationship with sine, cosine, and tangent. Using this relationship, we can find the exact values these functions!

##Finding Values

Let's look at an example right triangle with the sides labeled:

The sine of the angle indicated is $\frac{8}{17}$.

As we already studied...

$$\csc(\theta)= \frac{1}{\sin(\theta)}$$

Therefore, if $\sin(\theta) = \frac{8}{17}$, then $\csc(\theta) = \frac{1}{ \frac{8}{17}}$, which $=\frac{17}{18}$. In other words, to find values of reciprocal trig functions, we can find the reciprocal of it's corresponding sine, cosine, or tangent function.

For a more challenging problem, let's try finding the reciprocal function using unit circle values.

Reciprocal Trig Functions

We've learned about the individual reciprocal trigonometric functions, but let's put them all together!

We can also think of trigonometric reciprocal functions as:

$$sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} \rightarrow csc(x)=\frac{\text{hypotenuse}}{\text{opposite}}$$ $$cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} \rightarrow csc(x)=\frac{\text{hypotenuse}}{\text{adjacent}}$$ $$tan(x) = \frac{\text{opposite}}{\text{adjacent}} \rightarrow csc(x)=\frac{\text{adjacent}}{\text{opposite}}$$

If you know the length of the sides, you can fill in the given information and solve for what you are looking for! The pythagorean theorem is also an option, but you'll often find that if you know one angle, the reciprocal trig ratios are easier.

Solving Reciprocal Trig Functions

by Joel Speranza Math

In this video by Joel Speranza Math, you will learn how to solve for exact values of reciprocal trig functions.

Summary

Example 1

Solve for $\theta$: $\text{sec}(\theta) = -2$, $0<x<2\pi$

We know that secant is the inverse of cosine, so we can say that:

$$\begin{align} \frac{1}{\text{cos}(\theta)} &= -2 \\ -\frac{1}{2} &= \text{cos}(\theta) \\ \theta &= \frac{2\pi}{3}, \frac{4\pi}{3} \\ \end{align}$$

Try the next example on your own before watching the video!

Example 2

Solve for $\theta$: $\text{cot}(x) = 1$, $0<x<2\pi$