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