# What is the Inverse of Cosine?

## Inverse Functions

Hello, and welcome to this video on inverse trig functions! In order to understand what inverse trig functions are, let’s first review what normal trigonometric functions are. Remember, the common three trig functions are **sine**, **cosine**, and **tangent**. These trig functions are used to relate a triangle’s side and angle measures to one another. For instance, we would use tangent in a problem where we need to find the missing side length of a triangle.

If we remember SOH-CAH-TOA, we can see that this triangle uses TOA, which stands for the tangent of theta, which is our angle, is equal to the opposite over the adjacent.

\(tan\Theta =\frac{opposite}{adjacent} \)If we plug in the values for our triangle, we get the tangent of 30 is equal to 3 over x.

\(tan 30 =\frac{3}{x}\)Rearranging to get x by itself gives us that x equals 3 over the tangent of 30,

\(x=\frac{3}{tan 30}\)which then simplifies to x is approximately 5.20

x ≈ 5.20

But what if we are given a triangle where we know the side measures but we want to know the angle measure, like this.

If we plug it into our TOA equation, we get that the tangent of x equals 7 over 18.

\(tan x =\frac{7}{18}\)We want to get the x-value by itself, so we need to “undo” the tangent somehow. This is where inverse trig functions come in handy. Inverse trig functions are just the opposite of trig functions. The inverse of tangent is written as arctan x (which can look like atan x) or tan to the negative 1 power x, or tan inverse x. tan-1 x. Sine and cosine work the same way; just replace “tangent” with either “sine” or “cosine.”

If we apply this to our example, we get:

Now, all we have to do is plug this into our calculator and then we have our answer! The inverse trig functions are typically found by hitting the 2nd key and then the trig function key. If we do this, we find that x 21.25 degrees.

Let’s try another example. Use inverse trig functions to solve for x.

We are looking for our angle and we are given the opposite and hypotenuse side measures, so we are going to use SOH. Sine of x equals 3 over 6

\(sin x =\frac{3}{6}\)If we isolate x by itself, we get that x equals sine inverse of 3 over 6.

\(x=\frac{3}{6}\)Remember asin x and sine inverse mean the same thing, so the notation can be used interchangeably.

When we plug this into our calculator, we see that x = 30 degrees.

I want you to try one more, but this time do it on your own. After I show you the problem, pause the video and work it out. Then when you finish, see if your answer matches up with mine.

Solve for x:

In this problem, we are given our adjacent and hypotenuse sides and we are looking for the angle between them. This means we are going to use CAH. If we set up our equation, it will look like this:

\(cos x =\frac{9}{22}\)Solving for x by itself gives us x is equal to cosine inverse of 9 over 22.

\(x =\frac{9}{22}\)When we plug this into our calculator, we get that x is about 65.85 degrees.

Remember, inverse trig functions are just the opposite of trig functions. Trig functions are used to find the ratio of the sides of a triangle as related to the angle, and inverse trig functions help you figure out what that angle measure is when given the ratio of the sides.

I hope this review on inverse trig functions was helpful. Thanks for watching, and happy studying!