Hey, everyone, and welcome back. So up to this point, we've been spending a lot of time talking about trigonometric functions and how they relate to the sides of a right triangle. But something we haven't really covered yet is how we can deal with finding a missing angle for a right triangle. And that's what we're going to talk about in this video, because you can actually find these missing angles using something known as inverse trigonometric functions. So without further ado, let's learn about these operations that allow us to find these missing angles.
Now, for some problems that you're going to be given, you will have the value of a trigonometric function. And when this happens, it's actually pretty easy to find the missing angle because all you need to do this is the inverse of whatever trig function you have. To understand this concept of an inverse, we need to look at something we're a little more familiar with. For example, exponents. Let's say we have this number 3, and let's say I were to take this 3 and raise it to an exponent of 2. This would give us 32, and 32 = 9. But is there a way I can undo or reverse this operation that we just did? And the answer is yes, I can. I can take the square root, because the square root undoes the square, meaning that 9 will get us back to 3. So that's the idea of an inverse. The inverse undoes the operation you started with.
Now, when it comes to trigonometry, we also have inverse operations, known as inverse trig functions. Let's say that we have a 30-degree angle, and we take the sine of this angle. The sine of 30 degrees will give us 1/2. But is there a way to go backward and get 30 degrees from 1/2? And the answer is yes. What you need to do is take the inverse sine, because if you take the inverse sine of 1/2, this will cancel the sine operation that you took, meaning you're going to end up with your angle, which is 30 degrees. So that's the idea of an inverse trig function. It undoes the trig operation that you started with. Now something else I want to mention is these inverse trigonometric functions are also called arc functions. So if you see the inverse sine of theta, this is the exact same thing as the arcsine of theta. This is something to be familiar with in case you come across these arc functions. These are literally just the same thing as inverse functions.
Now, to make sure that we understand these concepts, let's actually try an example where we need to find the missing angle of a triangle. Here we have this triangle, and we're asked to write a trigonometric function to represent the angle theta. Let's see how we can do this, and we're going to go by the steps. Our first step when solving this problem is going to be to choose a trigonometric function, which includes the correct angle and sides for this triangle. The way that I'm going to do this is using our memory tool, SOHCAHTOA, because SOHCAHTOA tells us how the 3 main trig functions relate to the sides of a right triangle. What I notice is that we have our angle theta, and with respect to this angle, we have the opposite side of the triangle, as well as the hypotenuse. Looking at these options that we have, it looks like the sine includes the opposite and hypotenuse. So I'm going to use the sine of theta, and that is going to be our first step.
Now our next step is going to be to write an equation with the chosen trigonometric function. And to write an equation, recall that sine is opposite over hypotenuse. So if I go to our angle theta, the opposite side with respect to this angle is 6, and the hypotenuse or the long side is 13. So this would be the equation for the sine of theta that we're looking for, and that's step 2. Now, for step 3, we're asked to take the inverse on both sides of the equation to isolate the angle. To do this step, what I'm going to do is recognize we have the sine of theta. So, I'm going to take the inverse sine on both sides of this equation. θ = sin-1613. This right here is going to be our angle theta represented as this trigonometric function. This right here is the answer to this problem and what we're looking for for the angle theta.
In future videos, we're actually going to learn how you can take these inverse trigonometric functions and how you can plug them into a calculator to get an approximate value for this angle. But for now, this is all we're asked to find for this example, so that is the solution. Hopefully, you found this video helpful. This was a basic introduction to inverse trigonometric functions and finding missing angles of a right triangle. Let me know if you have any questions. Thanks for watching.