Sine and cosine of complementary angles
So we've got a right triangle drawn over here where this base's length is a, the height here is b, and the length of the hypotenuse is c. And we already know when we see something like this, we know from the Pythagorean theorem, the relationship between a, b, and c, we know there's a squared plus b squared is going to be equal to the hypotenuse squared, is going to be equal to c squared. What I want to do in this video is explore how we can relate trig functions to, essentially, the Pythagorean theorem. And to do that, let's pick one of these non-right angles. So let's pick this angle right over here as theta, and let's just think about this what the sine of theta is and what the cosine of theta is, and see if we can mess with them a little bit to somehow leverage the Pythagorean theorem. So before we do that, let's just write down sohcahtoa just so we remember the definitions of these trig functions. So sine is opposite over hypotenuse. Cah, cosine is adjacent over hypotenuse. And toa, tan is opposite over adjacent, we won't be using tan, at least in this video. So let's think about sine of theta. I will do it, I'll do it in this blue color. So sine of theta is what? It is opposite over hypotenuse, so it is equal to the length of b or it is equal to b-- b is the length-- b over the length of the hypotenuse, which is c. Now what is cosine of theta? Well, the adjacent side, the side of this angle that is not the hypotenuse, it has length a. So it's the length of the adjacent side over the length of the hypotenuse. Now how could I relate these things? Well it seems like, if I square sine of theta, then I'm going to have sine squared theta is equal to b squared over c squared, and cosine squared theta is going to be a squared over c squared. Seems like I might be able to add them to get something that's pretty close to the Pythagorean theorem here. So let's try that out. So sine squared theta is equal to b squared over c squared. I just squared both sides. Cosine squared theta is equal to a squared over c squared. So what's this sum? What's sine squared theta plus cosine squared theta? Is going to be equal to what? Sine squared theta is b squared over c squared, plus a squared over c squared, which is going to be equal to-- Well we have a common denominator of c squared. And the numerator, we have b squared plus a squared. Now, what is b squared plus a squared? Well, we have it right over here, Pythagorean theorem tells us, b squared plus a squared or a squared plus b squared is going to be equal to c squared. So this numerator simplifies to c squared. And the whole expression is c squared over c squared, which is just equal to 1. So using the sohcahtoa definition, in a future video, we'll use the unit circle definition. But you see just using the units, just even using the sohcahtoa definition of our trig functions, we see probably the most important of all the trig identities. That the sine squared theta, sine squared of an angle, plus the cosine squared of that same angle-- I'm introducing orange unnecessarily-- is going to be equal to 1. Now you might probably be saying, OK Sal, that's kind of cool, but what's the big deal about this? Why should I care about this? Well, the big deal is now you give me the sine of an angle and I can solve this equation for the cosine of that angle, or vice versa. So this is actually a pretty powerful, powerful thing. And this is also part of the motivation even for the unit circle definition of trig functions.