Hey, everyone, and welcome back. So up to this point, we've spent a lot of time talking about trig functions and right triangles, as well as some of these special case right triangles that allow us to solve problems super fast. Well, in this video, we're going to be learning about how we can find the missing side lengths for any kind of right triangle that we have, as long as we are given one side and one angle. Now this is a very important skill to have because you're not always going to have these special cases where you can use shortcuts. You're going to need to know how we can use the trigonometric functions and equations we've already learned about to solve any kind of right triangle. So without further ado, let's get right into this.

Now what we're going to do is take a look at this example, where we're asked to find all side lengths of the given triangle. Our first step should be to find any missing angles in this triangle because this gives us some nice options when we use our trigonometric functions. Now looking at the right triangle that we're given, I see that we have one side, which is the hypotenuse, and then we have this angle, which is 37 degrees, and nothing else is given to us. Now if we want to find this other non-right angle, what we can do is take the angle that we have here, and we can subtract it from 90 degrees. So if I take 90 degrees minus our angle a, which we'll say that our angle a is 37 degrees, we'll have 90 minus 37, which is 53. So that means that our missing non-right angle that we have there is going to be 53 degrees, and that's our first step.

Now our next step is going to be to choose a trigonometric function that includes one of the missing sides and the given side. Now what I'm going to do is see if I can find this missing side of the triangle, and this missing side is opposite our 37-degree angle. So I'm going to use this angle to see if I can find this missing side. Now the way that I can do this is by using the SOHCAHTOA memory tool. Recall that SOHCAHTOA tells us how the trigonometric functions relate to the sides of the right triangle. Now because we have the hypotenuse or the long side of the triangle, I can use either the sine or the cosine to find this missing side. But what I'm going to do is use the sine because we're going to have that the sine of our angle theta is equal to the opposite divided by the hypotenuse. And since I'm trying to find this missing side of the triangle, I'm going to use the angle that is opposite that side. So we're going to have that the sine of our 37-degree angle is equal to the opposite side of this triangle, and the side opposite is the missing side x divided by the hypotenuse, which in this case is 5.

So now that we have this equation set up, our third step is going to be to solve for x, which is the second side length of the triangle. What I'm first going to do is multiply both sides of this equation by 5, and that's going to get the fives to cancel on the right side. Notice how that leaves us with just x by itself. And we'll have that x is equal to 5 times the sine of 37 degrees. So all you need to do is take 5 times the sine of 37 and plug it into your calculator. And make sure your calculator is in degree mode. If you do this, you should get an approximate answer of 3. Now, on your calculator, it's going to read as, like, 3.009075, and it's going to keep going on. But we can say that that's about equal to 3. So that means that our missing side x is 3.

Now our last step is going to be to find the final missing side using the Pythagorean theorem. And recall that this is what the Pythagorean theorem looks like. So the way that I can do this is by recognizing that we have 2 of the sides, and we're missing this third side, which I'm going to call b. I'm going to say that this side we just calculated is a, and then the hypotenuse is always equal to c. So we have that a squared plus b squared equals c squared. So we can say that 3 squared plus b squared is equal to c or 5 squared. So 3 squared is equal to 9, and then we have plus b squared, and that's equal to 5 squared, which is 25.

Now from here what I can do is take 9 and subtract it on both sides of this equation. This is going to get the nines to cancel on the left side, leaving us with just b^{2}. And the last b^{2} is equal to 25 minus 9, which is 16. Now our last step is going to be to take the square root on both sides of this equation, canceling the square on the b, leaving us with just b. And we'll have that b is equal to the square root of 16, which is 4. So that means that our missing side b is 4. And notice how we were able to use trigonometric functions and the Pythagorean theorem to find all missing sides of this right triangle. So this is the strategy you can use to solve any kind of right triangle, as long as you are given one side and one angle. I hope you found this video helpful. Thanks for watching.