WEBVTT
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[MUSIC PLAYING]
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Hello.
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It's your turn on trigonometry
by the unit circle.
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OK.
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Now let us recall some
laws that will permit
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us to reduce our computations to
angles between 0 and pi over 2.
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OK.
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The laws that I
wish to remind you
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are the antipodal law, which
says that the sine of pi plus x
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is equal to minus the sine of
x, and the cosine of pi plus x
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is equal to minus
the cosine of x.
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Then we have another
important fact,
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that the sine is
an odd function.
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That is, the sine of minus x is
equal to minus the sine of x.
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And the cosine is
an even function.
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That is, the cosine of minus
x is equal to the cosine of x.
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And finally, we
remember the periodicity
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of these trigonometric
functions.
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That is, the sine of x plus any
integer multiple of 2 times pi
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is equal to sine of x.
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And the cosine of x plus
any integer multiple of 2 pi
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is equal the cosine of x.
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OK.
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Now with these laws, let
us see that it's very
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easy to compute these values.
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Good.
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We will compute these
values applying these laws.
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You will see that when you
have more practice, then
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you can maybe follow
other quicker procedures.
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OK.
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But let us apply these laws.
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First of all, the
cosine of 2 pi over 3,
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I want to reduce
this computation
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to an angle between
0 and pi over 2.
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Good.
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I observe that this is equal
to the cosine of pi plus minus
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pi over 3, because this
is pi minus pi over 3.
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And then applying
the antipodal law,
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I get that this is equal to
minus the cosine of minus pi
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over 3.
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OK.
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But the cosine is
an even function.
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Therefore, this is equal to
minus the cosine of pi over 3.
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And now the cosine of pi
over 3 is equal to 1 over 2,
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therefore, minus the
cosine is minus 1 over 2.
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good.
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And now, let us consider the
tangent of minus pi over 3.
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OK.
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By definition, this is equal
to the sine of minus pi over 3
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over the cosine of
minus pi over 3.
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OK.
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Because the sine
is an odd function,
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this is equal to minus the
sine of pi over 3 over--
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now, the cosine is
an even function.
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And therefore, this is equal
to the cosine of pi over 3.
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Good.
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We have reduced ourself to
angles between 0 and pi over 2.
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And now, the sine
of pi over 3 is
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equal to the square
root of 3 over 2.
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Therefore, at the numerator
we get minus the square root
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of 3 over 2.
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And at the denominator,
the cosine of pi over 3
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is 1 over 2.
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And then we get minus
the square root of 3.
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Good.
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And finally now,
we have to compute
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the sine of 19 pi over 4.
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OK.
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Clearly, this angle is
greater than 2 times pi.
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Therefore, we can
apply the periodicity
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to reduce this angle.
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What do we get?
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Observe that 19 pi
over 4 is equal to 16
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pi plus 3 pi over 4, which
is the sine of 4 times pi
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plus 3 pi over 4.
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Good.
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Now, this is a multiple of 2 pi.
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Therefore, this is equal
to the sine of 3 pi over 4.
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OK.
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And now this is
the sine of what?
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OK.
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This is exactly like
pi minus pi over 4.
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That is, pi plus
minus pi over 4.
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And by the antipodal laws,
this is equal to what?
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To the minus sine
of minus pi over 4.
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OK.
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But the sine is an odd function.
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Therefore, the sine
of minus pi over 4
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is minus the sine of pi over 4.
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We have another minus here.
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And therefore, we get
the sine of pi over 4.
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OK.
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We have reduced ourselves to an
angle between 0 and pi over 2.
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And the sine of pi over 4
is equal to the square root
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of 2 over 2.
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OK.
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Thank you very much
for your attention.
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