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# Using complementary angles

Video transcript

We are told that the
cosine of 58 degrees is roughly equal to 0.53. And that's roughly equal
to, because it just keeps going on and on. I just rounded it to
the nearest hundredth. And then we're asked, what
is the sine of 32 degrees? And I encourage you
to pause this video and try it on your own. And a hint is to look
at this right triangle. One of the angles is
already labeled 32 degrees. Figure out what
all of the angles are, and then use the
fundamental definitions, your sohcahtoa definitions,
to see if you can figure out what sine of 32 degrees is. So I'm assuming you've
given a go at it. Let's work it through now. So we know that the sum of
the angles of a triangle add up to 180. Now in a right angle, one
of the angles is 90 degrees. So that means that the
other two must add up to 90. These two add up to
90 plus another 90 is going to be 180 degrees. Or another way to think
about is that the other two non-right angles are
going to be complementary. So what plus 32 is equal to 90? Well, 90 minus 32 is 58. So this right over here
is going to be 58 degrees. Well, why is that interesting? Well, we already know what
the cosine of 58 degrees is equal to. But let's think
about it in terms of ratios of the lengths of
sides of this right triangle. Let's just write down sohcahtoa. Soh, sine, is opposite
over hypotenuse. Cah, cosine, is adjacent
over hypotenuse. Toa, tangent, is
opposite over adjacent. So we could write
down the cosine of 58 degrees, which
we already know. If we think about it in terms
of these fundamental ratios, cosine is adjacent
over hypotenuse. This is a 58 degree angle. The side that is adjacent
to it is-- let me do it in this color-- is side
BC right over here. It's one of the sides
of the angle, the side of the angle that is
not the hypotenuse. The other side, this over
here, is a hypotenuse. So this is going to be
the adjacent, the length of the adjacent side, BC, over
the length of the hypotenuse. The length of the
hypotenuse, well, that is AB. Now let's think about what the
sine of 32 degrees would be. Well, sine is opposite
over hypotenuse. So now we're looking at
this 32 degree angle. What side is opposite it? Well, it opens up onto BC. And what's the length
of the hypotenuse? It's AB. Notice, the sine of 32
degrees is BC over AB. The cosine of 58
degrees is BC over AB. Or another way of thinking
about it, the sine of this angle is the same thing as the
cosine of this angle. So we could literally
write the sine-- I want to do that in that pink
color-- the sine of 32 degrees is equal to the cosine
of 58 degrees, which is roughly equal to 0.53. And this is a really,
really useful property. The sine of an angle is equal
to the cosine of its complement. So we could write
this in general terms. We could write that
the sine of some angle is equal to the cosine
of its complement, is equal to the cosine
of 90 minus theta. Think about it. I could change this
entire problem. Instead of making this
the sine of 32 degrees, I could make this the
sine of 25 degrees. And if someone gave you the
cosine of-- what's 90 minus 25?-- if someone gave you
the cosine of 65 degrees, then you could think
about this as 25. The complement is going
to be right over here. This would be 65 degrees. And then you could use
the exact same idea.