 In a previous video, we demonstrated the so-called reciprocal and quotient identities of the six trigonometric ratios. It shows how the functions are related to each other. Co-secand is just the reciprocal of sine. Tangent is just sine divided by cosine. It turns out there is a litany of relationships between trigonometric functions. In this video, I want to talk about the complementary relationships between them. That is, why do we call some of them cosine and co-tangent, co-secand? Why the co? Well, co is actually short for complementary in this context. And so in the language of trigonometry, we say that sine and cosine are co-functions, are complementary functions, co-functions just for short. So sine and cosine are co-functions of each other. Tangent and co-tangent are co-functions of each other. And secant and co-secant are complementary functions of each other. So every trigonometric function comes with a complement. Sine is the complement of cosine and vice versa. Co-tangent is the complement of tangent and vice versa. And then secant and co-secand are complements of each other. And so the reason why we call them complements of each other actually comes from the word complementary angles. If we were to take a right triangle like this one, ABC, because the sum of the angle measures in a triangle is always gonna add up to be 180 degrees, if one of the angles C is 90 degrees, then the other two have to add up to be 90 degrees so that the total sum is 180 degrees. So the two non-right angles of a right triangle are always complementary angles, right? A plus B will always add up to be 90 degrees. And so if you take, depending on your perspective, will affect the trigonometric ratios. Notice the following, if I look at the angle A and I ask myself, what is say cosine of A? Cosine of A here, this is gonna be the adjacent side over the hypotenuse, so you get B over A. On the other hand, if I were to look at the angle B and I ask myself, what's cosine of B? Well, the role of who's the adjacent side changes, right? So B thinks that little A is the adjacent side. The hypotenuse doesn't depend on the perspective there, right? On the other hand, if I do sine for A, sine of A is gonna give you opposite over hypotenuse. You get A over C. And if I were to do sine of B, again, the opposite side will be B over C, like so. And I wanna point out to you that cosine of A is the same thing as sine of B. And likewise, cosine of B is the same thing as sine of A. That is, if you switch from a function, excuse me, if you switch from an angle to its complementary angle, you're gonna switch from a trigonometric function to its complementary trigonometric function. Because again, the role of who's the opposite side and who's the adjacent side switches as you go from one angle to its complement. So we see that sine of B is the same thing as cosine of A. Cosine of B is the same thing as sine of A. Tangent of B, you can run through that one, that's gonna be tangent, cotangent of A, right? Because what is tangent of B? Tangent of B is the opposite over adjacent, opposite with relative to B, adjacent relative to B, that's gonna be little B over A. Cotangent on the other hand, that's gonna be the adjacent of A divided by the opposite of A. But for A, the adjacent side is B and the opposite side is A. The roles are reversed. And but you get the same ratio. Tangent of B is equal to cotangent of A. Cotangent of B is equal to tangent of A for the same reasons. If I already consider secant of B, what is secant of B? Secant, you take the hypotenuse divided by the adjacent side, adjacent relative to B. Well, the hypotenuse is always gonna be C. The adjacent side to angle B is side A. On the other hand, cosecant of A, you take hypotenuse divided by the opposite side. Relative to the angle A, the opposite side is little A. And so you get the same ratio of C over A. Similarly, cosecant of B is equal to secant of A. And so whatever you switch from an angle to its complement, you switch the trig function to also to its complement. And so this is called the co-function theorem. A trigometric function of an angle is always equal to the co-function of the complement of the angle. So this means like sine of B equals cosine of A when A and B are complementary angles. So some specific examples of this. The co-function theorem tells us that sine of 30 degrees will equal cosine of 60 degrees. You'll notice that 30 and 60 are complementary angles and sine and cosine are complementary functions. Likewise, tangent of 75 degrees will be the same as cotangent of 15 degrees. Why is that? Well, 75 and 15 are complementary angles. Their sum adds up to 90 degrees and tangent and cotangent are complementary functions. Likewise, if I take secant of theta, this will equal cosecant of 90 degrees minus theta. That is whatever theta turns out to be, if I switch it to its complement, that'll turn it into a cosecant. And so this right here are so-called complementary identities. Sine of theta is equal to cosine of 90 degrees minus theta. We can do tangent of theta equals cotangent of 90 degrees minus theta. And you can repeat this for all six trigometric functions. If you switch to the complement of the angle, you'll also switch the function to its complement as well. Consider, this is where we'll end this video. Consider the equation cosine of theta plus four degrees equals sine of three theta plus two degrees. Let's solve this equation. Now, I wanna warn you that as we solve this equation, I'm not gonna give you all of the solutions to this equation. Because it turns out there's a little bit more going on when it comes to solving trigonometric equations. This is a topical tackle in much more detail later in this lecture series. But just to demonstrate the co-function theorem, notice that if cosine and sine are equal to each other, these are complementary functions. So if the angles are complementary, that'll give us a solution. So for example, if I take theta plus four degrees, if we take that angle plus three theta plus two degrees, like so, oh, let me move those. The degrees should, we're saying these are all degrees here. This should add up to 90 degrees, right? So the sum of the two angles, if the two angles are complementary, that'll be a solution to this equation right here. And so adding like terms, you're gonna get theta plus three theta, which is a four theta. You're gonna get four plus two, which is a six that equals 90 degrees. Subtract six from both sides. You get four theta equals 84. And divide both sides by four, you're gonna get theta equals 21 degrees. So it turns out that 21 degrees is a solution to this equation, right? And like I said, yeah, you have to be careful because it turns out 21's not the only solution. In terms of considering the period of sine and cosine, you also get 111 degrees, 201 degrees, and then 291 degrees also solves this thing. And it turns out that there is sort of an argument one has to use with reference angles because sine and, since sine in the first quadrant and the second quadrant will have the same value, turns out there's also solutions of 46 and 226. But like I said, the main idea here was that the co-function theorem gives you 21 degrees as a solution. And then using other techniques we haven't introduced yet in the series, we could derive these other solutions between zero and 360 degrees.