Defined and Reciprocal Functions

Hi, and welcome to this video on reciprocal trig functions!

Before we dive into what reciprocal trig functions are, let’s quickly review what normal trig functions are.

Common Trig Functions

The three most common trig functions are sine, cosine, and tangent. When given a right triangle, sine is the ratio of the triangle’s opposite to its hypotenuse, cosine is the ratio of its adjacent to its hypotenuse, and tangent is the ratio of its opposite to its adjacent. We remember these by using the phrase “SOH-CAH-TOA”.

When you take the reciprocal of an expression, you divide one by the expression. A super simple way of doing this is to turn your expression into a fraction and then flip it. So if our expression is 2, we would turn that into 2 over 1, which makes it easy to see that the reciprocal is one half. The reciprocal of three fifths is five thirds. And the reciprocal of $2x$ is $\frac{1}{2x}$.

Reciprocal Functions

With reciprocal trig functions, we have the same thing, except the reciprocals are given new names.

Cosecant

Let’s start with the reciprocal of sine, cosecant. Cosecant is the same as 1 over sine, or when you are given a triangle, the ratio of the triangle’s hypotenuse to its opposite. This is just the flipped version of SOH, sine equals opposite over hypotenuse.

Take a look at this example.

Given this triangle, what is the cosecant of theta?

First, we need to remember that cosecant is the reciprocal of sine, so that means we need to figure out what sine is and flip it. Sine is opposite over hypotenuse, so in this case, $\frac{5}{13}$. Then we flip that to get cosecant of theta, which is $\frac{13}{5}$.

Secant

The second reciprocal function we are going to look at is secant, the reciprocal of cosine. Secant is the ratio of a triangle’s hypotenuse to its adjacent, or $\frac{H}{A}$, which, if you think back to our CAH from SOH-CAH-TOA is the flipped version of cosine equals $\frac{A}{H}$.

Let’s try an example problem using secant.

Find the missing side using secant.

If we set up our equation, we get $\text{sec}47=\frac{29}{x}$ since our angle is measured 47 degrees and our hypotenuse is 29.

This can be rearranged so that $x$ is by itself, which looks like this: $x=\frac{29}{\text{sec}47}$.

When we plug this into our calculator, making sure our calculator is in degrees mode instead of radians, we get that $x$ is approximately 19.78.

Cotangent

We have one more reciprocal function to cover and that is the reciprocal of tangent, which is cotangent. Cotangent is 1 over tangent, or the ratio of a triangle’s adjacent over its hypotenuse.

Using the things we have learned from our examples on cosecant and secant, I want to try another one, this time using cotangent.

Given this triangle, what is the cotangent of theta?

We know that tangent is opposite over adjacent, so since cotangent is the reciprocal of tangent, it must be the adjacent over opposite. This gives us the ratio $\frac{12}{9}$, so the cotangent of theta is $\frac{12}{9}$.

Importance of Knowing Reciprocals

Now that we have covered all the reciprocal trig functions, I want to mention something important about them. Reciprocal trig functions pop up all the time in homework or test problems, so it is important to recognize the names and recall what function they go to. However, as you may have noticed from our examples, when solving triangles by yourself, you almost always are able to use the original, or defined, trig function instead of its reciprocal.

Most commonly you will only use sine, cosine, and tangent, but as I said, the reciprocal trig functions will come up on homework and tests occasionally, so it is very important to recognize them and understand what they mean even if you do not use them in your own personal application of trigonometry.

Example Problem

Before we end this video, I want to give you one more example to try on your own. Once I put it up, pause the video and see what answers you can come up with. Then you can check them with mine and see how you did.

Given this triangle, find the cosecant of theta, the secant of theta, and the cotangent of theta.

Remember, cosecant is the reciprocal of sine so we are looking for the hypotenuse over the opposite. The cosecant of theta is $\frac{25}{7}$. Secant is the reciprocal of cosine, so we need the hypotenuse over the adjacent, which gives us that secant of theta is equal to $\frac{25}{24}$. And finally, cotangent is the reciprocal of tangent so we need the adjacent over the opposite, which leaves us with cotangent of theta is $\frac{24}{7}$.

I hope this review of reciprocal trig functions was helpful. Thanks for watching and happy studying!

Defined and Reciprocal Function Practice Questions