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# Sine & cosine of complementary angles

Sal shows that the sine of any angle is equal to the cosine of its complementary angle. Created by Sal Khan.

Video transcript

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.