Sine & cosine of complementary angles

We see in a triangle, or I guess we know in a triangle, there's three angles. And if we're talking about a right triangle, like the one that I've drawn here, one of them is going to be a right angle. And so we have two other angles to deal with. And what I want to explore in this video is the relationship between the sine of one of these angles and the cosine of the other, the cosine of one of these angles and the sine of the other. So to do that, let's just say that this angle-- I guess we could call it angle A-- let's say it's equal to theta. If this is equal to theta, if it's measure is equal to theta degrees, say, what is the measure of angle B going to be? Well, the thing that will jump out at you-- and we've looked at this in other problems-- is the sum of the angles of a triangle are going to be 180 degrees. And this one right over here, it's a right triangle. So this right angle takes up 90 of those 180 degrees. So you have 90 degrees left. So these two are going to have to add up to 90 degrees. This one and this one, angle A and angle B, are going to be complements of each other. They're going to be complementary. Or another way of thinking about it is B could be written as 90 minus theta. If you add theta to 90 degrees minus theta, you're going to get 90 degrees. Now, why is this interesting? Well, let's think about what the sine of theta is equal to. Sine is opposite over hypotenuse. The opposite side is BC. So this is going to be the length of BC over the hypotenuse. The hypotenuse is side AB. So the length of BC over the length of AB. Now, what is that ratio if we were to look at this angle right over here? Well, for angle B, BC is the adjacent side, and AB is the hypotenuse. From angle B's perspective, this is the adjacent over the hypotenuse. Now, what trig ratio is adjacent over hypotenuse? Well, that's cosine. Sohcahtoa, let me write that down. Doesn't hurt. Sine is opposite over hypotenuse. We see that right over there. Cosine is adjacent over hypotenuse, cah. And toa, tangent is opposite over adjacent. So from this angle's perspective, taking the length of BC, BC is its adjacent side, and the hypotenuse is still AB. So from this angle's perspective, this is adjacent over hypotenuse. Or another way of thinking about it, it's the cosine of this angle. So that's going to be equal to the cosine of 90 degrees minus theta. That's a pretty neat relationship. The sine of an angle is equal to the cosine of its complement. So one way to think about it, the sine of-- we could just pick any arbitrary angle-- let's say, the sine of 60 degrees is going to be equal to the cosine of what? And I encourage you to pause the video and think about it. Well, it's going to be the cosine of 90 minus 60. It's going to be the cosine of 30 degrees. 30 plus 60 is 90. And of course, you could go the other way around. We could think about the cosine of theta. The cosine of theta is going to be equal to the adjacent side to theta, to angle A, I should say. And so the adjacent side is right over here. That's AC. So it's going to be AC over the hypotenuse, adjacent over hypotenuse. The hypotenuse is AB. But what is this ratio from angle B's point of view? Well, the sine of angle B is going to be its opposite side, AC, over the hypotenuse, AB. So this right over here, from angle B's perspective, this is angle B's sine. So this is equal to the sine of 90 degrees minus theta. So the cosine of an angle is equal to the sine of its complement. The sine of an angle is equal to the cosine of its complement.