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 talk 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 minus theta you're going to get you're going to get or theta to 90 degrees minus theta you're going to get 90 degrees now why is this interesting well let's think about let's think about what the sine 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 a B so the length of BC over the length of a B over the length of a B 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 a B is the hypotenuse for 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 sine doesn't hurt sine is opposite over just we see that right over there cosine is adjacent over hypotenuse huh and Toa tangent is opposite over adjacent so from this angles perspective taking the length of BC which is BC is its adjacent side and the hypotenuse is still a B so from this angles perspective this is adjacent over hypotenuse or another way of thinking but it's the cosine of this angle so that's going to be equal to the cosine cosine of 90 degrees minus 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 one 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 this 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 a B but what is this ratio from angle B's point of view well the sine of angle B the sine of angle B is going to be its opposite side AC AC over the hypotenuse a B so this right over here from angle B's perspective this is angle B's sine so this is equal to this is equal to the sine of 90 minus 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