Hey, everyone, and welcome back. So in earlier videos, we've brushed up on this idea of angles and triangles. What we're going to be looking at in this video is arguably one of the most critical topics both in this course and in future math courses that you will likely take, which is the idea of trigonometric functions, which we also call trig functions for short. Now, if you're not sure what trig functions are, all these do is they relate angles to the side lengths of a right triangle. In this video, there are three main types of trig functions we're going to talk about, which are the sine, cosine, and tangent. That might sound a little confusing or like it's a bunch of gibberish, but don't sweat it because all we're going to learn is that these three trigonometric functions are just ratios. An example of a ratio would be something like 3 fifths or 4 fifths. So these types of fractions or ratios are all that these trigonometric functions are. Without further ado, let's get into this video and see how we can solve some problems where we need to find these specific ratios based on a right triangle. Now, how can we go about finding the sine of some angle in this triangle that we'll call theta? To find the sine of this, sine is opposite over hypotenuse. So if you wanted to find the sine of this angle theta, what you do is look at the angle that you have and go to the opposite side of the triangle, which in this case is 3, and then divide that by the hypotenuse, which is always the long side of the triangle or 5. So 3 fifths is going to be the sine of theta. As you can see, solving for these trigonometric functions is pretty straightforward.

So let's now say we wanted to find the cosine of theta. Well, if we're dealing with the cosine this time, what we need to do is look at our angle theta and find the adjacent side and divide it by the hypotenuse. The adjacent side of this triangle is the side next to the angle which in this case is 4, and then this is divided by the hypotenuse or long side of the triangle which is 5. So the cosine of our angle theta is 4 fifths. But now let's say we wanted to find the tangent of our angle theta. If we're dealing with the tangent this time, what we need to do is take the opposite side of the triangle and divide it by the adjacent side. So the opposite side to our angle theta is 3, and then the adjacent side is 4. So 3 fourths would be the tangent of our angle theta. Now, you might feel like this is a lot to remember. Like, how are we just supposed to know that sine is opposite over hypotenuse, or that tangent is opposite over adjacent? Well, it turns out there is a memory tool that we use to remember this, and that is SOH CAH TOA. This memory tool, SOH CAH TOA, tells us how each of the trigonometric functions relates to the sides of the right triangle. So we have that sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. One more thing I want to mention about the tangent is that the tangent of theta is also equal to the sine of theta divided by the cosine of theta. So if we were to take the sine of theta, which we figured out was 3 fifths, and divide that by the cosine of theta, which we figured out was 4 fifths, notice how the fives would just cancel in this fraction, so all we would end up with is 3 over 4, and 3 over 4 is exactly what we calculated the tangent to be. So this is another relationship to be familiar with when dealing with these trigonometric functions.

Now to really make sure we have these trig functions and this SOHCAHTOA memory tool down, let's try some examples where we have to solve some problems based on a right triangle. Now, in this problem, we are asked to find the values of the trig function indicated given the right triangle that we have over here. We're going to start with example a and before I do anything what I'm going to do is write down the memory tool SOHCAHTOA. Now looking at example a we have the sine And sine is opposite over hypotenuse according to SOHCAHTOA. So we'll have the opposite side of our right triangle divided by the hypotenuse. Now we're dealing with angle x in this example, so if we go to angle x in our triangle, we're going to have the side opposite to angle x, which is 12, divided by the hypotenuse or the long side, which is 13. So 12 over 13 is the ratio we're looking for and the answer for example a. Now let's try solving example b where we are asked to find the tangent of x. Well, if we once again go to our angle x here, we can find the tangent by looking at sohcahtoa, which says that tangent is opposite over adjacent. So if we take this ratio, we can go to our angle x, we can find the side opposite to x, which in this case is 12, and then divide it by the adjacent side or the side next to x, which in this case is 5. So 12 over 5 is the ratio that we're looking for for example b. But now let's try solving example c, where we are asked to find the cosine of our angle y. When dealing with the cosine, cosine is adjacent over hypotenuse based on this memory tool SOHCAHTOA. What you might think we need to do is look at our angle and then find the adjacent side which would be 5 and divide it by the hypotenuse or the long side which is 13. Now if 5 13ths is what you're thinking the correct ratio is, this is actually not correct, and here's the reason why. Notice in this example, we are dealing with angle y. And because we're dealing with angle y, we can no longer use x as a reference. We now have to use y as the reference angle. So rather than having 5 13ths, we're going to have the ratios based on the adjacent and hypotenuse of y. So if we go to y, the adjacent side to y is 12, and then the hypotenuse or the long side is still 13. So 12 over 13 would be the ratio for the cosine of y. So this is something you need to watch out for when dealing with these types of problems. Because notice in the first two examples, we were dealing with angle x, so we used x as a reference. Where in this example, we were dealing with y, so we used angle y as a reference. So that's the main idea behind solving trigonometric functions and using this SOHCAHTOA memory tool. Hope you found this video helpful. Thanks for watching, and please let me know if you have any questions.