Hey, everyone, and welcome back. Up to this point, we've spent a lot of time talking about trigonometric functions and how they relate to right triangles. Now, what we're going to learn about in this video is this new concept of cofunction identities as well as complementary angles. This might sound a bit confusing and like something that's completely different from what we've been learning about so far, but if that's how you're feeling, don't sweat it. Because what we're going to learn about in this video is that both of these concepts relate to these trigonometric functions that we've already been learning about throughout these videos. And I think you're going to find that both of these concepts are very straightforward and relate really nicely to this big concept of the right triangle. So without further ado, let's get right into things. To understand complementary angles, these are just angles that add to 90 degrees. One of the most basic examples of complementary angles would be the non-right angles in a right triangle because in a right triangle, these non-right angles are always complementary. To give an example of this, let's take a look at a right triangle down here. We know that one of these angles in the right triangle has to be 90 degrees because that's what a right triangle is. And because this angle has to be 90 degrees, all the angles in a triangle have to add to 180 degrees. So that means these two angles have to add to 90 degrees because 90 plus 90 would give you 180. So that's why these two non-right angles are always going to be complementary. The reason it's important to understand complementary angles is that this ties into something called the complementary angle theorem. The complementary angle theorem says that cofunctions of complementary angles are equal. To understand what cofunctions are, you need to know that we also call these cofunction identities, and cofunction identities describe how two trigonometric functions are related to each other. One of the most basic examples of this is the sine and the cosine. These are examples of cofunctions, because if you were to take the sine of some angle, let's say 53 degrees, you could say that this is equal to the cosine of the complementary angle. The complementary angle would just be 90 degrees minus whatever angle we started with, which in this case is 53 degrees. And 90 minus 53 comes out to 37 degrees, and that makes sense because we already said these angles were complementary. So the sine of 53 degrees is equal to the cosine of 37 degrees. The function of some angle is going to be equal to the cofunction of its complementary angle. And this works for every cofunction identity. For example, the cofunction to secant would be cosecant, and the cofunction to tangent would be cotangent. Notice for each of the cofunctions, all we do is put the word "co" in front of the original function, like secant and cosecant, or sine and cosine. That's an easy way you can remember what the cofunction is for the original function. So if we go back to this right triangle that we have and we have the sine of 37 degrees and we want to find what the cofunction is, we know the cofunction would be the cosine and all we need to do is find the complementary angle which would be 90 degrees minus whatever angle we started with in this case 37 degrees, and 90 minus 37 comes out to 53 degrees. The sine of 37 degrees is equal to the cosine of 53 degrees. And if you found the ratios for this triangle for the sine of 37 degrees and the cosine of 53 degrees, you'd find both of these come out to the fraction 3 5 . That's the main idea behind cofunction identities and complementary angles. To make sure we understand this concept, let's try some examples.

In these examples, we're asked to write the expression in terms of the appropriate cofunction.

Let's start with example a, where we're asked to find the cofunction for the sine of 30 degrees. We know that the cofunction for sine is cosine, and what we need to do is find the cosine of the complementary angle. Well, the complementary angle would just be our original angle subtracted from 90 degrees. So we would have 90 degrees minus our original angle, which is 30 degrees, and 90 minus 30 comes out to 60 degrees. The cosine of 60 degrees would be the answer to example a.

Now let's take a look at example b where we have the tangent of 16 degrees. Well, the cofunction for tangent is cotangent, and then we need to find the complementary angle which would be 90 degrees minus the original angle we had, which is 16 degrees. 90 minus 16 comes out to 74. So we'd have the cotangent of 74 degrees, and this would be the solution for example b.

Now let's take a look at example c where we have the secant of 0 degrees. Well, the cofunction for secant is cosecant, and this will be the cosecant of the complementary angle, which would be 90 degrees minus our original angle, which is 0 degrees. This might seem a little bit strange since we're starting with a 0-degree angle, but we can solve this the exact same way we've solved these other problems. So 90 minus 0 comes out to 90, so we would just have the cosecant of 90 degrees.

But now let's try example d. How can we go about solving this problem? This one's a little bit complicated because notice that we have a cofunction that we're starting with, we have cosine and then we have this pi in here. So this is going to be a little bit tricky. But we can solve this just like we've solved these other problems where we first find the cofunction for cosine. And as we've discussed up here, sine and cosine are cofunctions, so what we would need here is the sine, which is the cofunction of cosine. Now rather than having 90 degrees minus our original angle, we can't do that because since we have pi, that means we're dealing with radians rather than degrees. And the radian equivalent of 90 degrees is π 2 . So what that means is we need to have π 2 minus the original angle which in this case is 5 π 18 . So this is really going to be the cofunction for this example. Now to solve this example, what I can do is I can actually get like denominators, and I can do that by multiplying the top and bottom of this fraction on the left side by 9. This will leave us with sine, and we're going to have 9 π 18 minus 5 π 18 . Notice when doing this, we get the same denominator of 18 for both of these fractions. Now what I need to do from here is subtract 5 from 9 and then 5 minus 9 is 4, so we're going to have 4 and then times π because this is 9π minus 5π, and this is going to be all divided by 18. Now at this point, I can reduce this fraction because 4 is the same thing as 2 times 2, and 18 is the same thing as 2 times 9. I can cancel one of the twos in this fraction, meaning we're going to end up with the sine of 2 π 9 . So sine of 2 π 9 is the solution to example d, and that is how you can solve these types of problems involving cofunction identities. Hope you found this video helpful. Thanks for watching, and please let me know if you have any questions.