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Determine the six
trigonometric ratios for angle A in the
right triangle below. So this right over here is
angle A, it's at vertex A. And to help me remember
the definitions of the trig ratios-- and these are human
constructed definitions that have ended up being very,
very useful for analyzing a whole series of
things in the world. To help me remember them, I
use the words soh cah toa. Let me write that down. Soh cah toa. Sometimes you can think
of it as one word, but it's really the three parts
that define at least three of the trig functions for you. And then we can
get the other three by looking at the first three. So soh tells us that sine of an
angle-- in this case it's sine of A-- so sine of A is equal
to the opposite, that's the O, over the hypotenuse. Well in this context, what is
the opposite side to angle A? Well, we go across the triangle,
it opens up onto side BC. It has length 12. So that is the opposite side. So, this is going
to be equal to 12. And what's the hypotenuse? Well, the hypotenuse is the
longest side of the triangle. It's opposite the
90 degree angle. And so we go opposite
the 90 degree angle, longest side is side AB. It has length 13. So this right over
here is the hypotenuse. So, the sine of A is 12/13. Now let's go to cah. Cah defines cosine for us. It tells us that cosine of
an angle-- in this case, cosine of A-- is equal
to the adjacent side to the angle over
the hypotenuse. So, what's the adjacent
side to angle A? Well, if we look
at angle A, there are two sides that
are next to it. One of them is the hypotenuse. The other one has length 5. The adjacent one is side CA. So it's 5. And what is the hypotenuse? Well, we've already
figure that out. The hypotenuse is
right over here, it's opposite the
90 degree angle. It's the longest side
of the right triangle. It has length 13. So the cosine of A is 5/13. And let me label this. This right over here
is the adjacent side. And this is all
specific to angle A. The hypotenuse would be the
same regardless of what angle you pick, but the
opposite and the adjacent is dependent on
the angle that we choose in the right triangle. Now let's go to toa. Toa defines tangent for us. It tells us that the
tangent of an angle is equal to the opposite
side over the adjacent side. So given this definition,
what is the tangent of A? Well, the opposite side,
we already figured out, has length 12. And the adjacent side,
we already figure out, has length 5. So the tangent of A, which
is opposite over adjacent, is 12/5. Now, we'll go the to the
other three trig ratios, which you could think of
as the reciprocals of these right over here. But I'll define it. So first you have cosecant. And cosecant, it's always
a little bit unintuitive why cosecant is the
reciprocal of sine of A, even though it starts
with a co like cosine. But, cosecant is the
reciprocal of the sine of A. So sine of A is opposite
over hypotenuse. Cosecant of A is
hypotenuse over opposite. And so what's the hypotenuse
over the opposite? Well, the hypotenuse is 13
and the opposite side is 12. And notice that 13/12 is
the reciprocal of 12/13. Now, secant of A
is the reciprocal. So instead of being
adjacent over hypotenuse, which we got from the
cah part of soh cah toa, it's hypotenuse over adjacent. So what is the secant of A? Well, the hypotenuse, we've
figured out multiple times already, is 13. And what is the adjacent side? It's 5. So it's 13/5, which
is, once again, the reciprocal of the
cosine of A, 5/13. Finally, let's
get the cotangent. And the cotangent is the
reciprocal of tangent of A. Instead of being
opposite over adjacent, it is adjacent over opposite. So what is the cotangent of A? Well, we've figured
out the adjacent side multiple times for
angle A. It's length 5. And the opposite side
to angle A is 12. So it's 5/12, which is,
once again, the reciprocal of the tangent of
A, which is 12/5.