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NARRATOR: Exercise 3.

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Find the exact value of:
a, sin 210 degrees,

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b, cos of 315 degrees,

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and c, tan of 150 degrees.

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Part a.

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Find the exact value
of sin of 210 degrees.

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So we first establish where
in the unit circle 210 degrees lies,

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and it lies in the third quadrant.

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So the coordinates of this point

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where this line of 210 degrees
meets the unit circle,

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this point has coordinates
(cos 210 degrees, sin 210 degrees).

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And we note that this line
is 30 degrees away from the x-axis

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so we should be able to relate these
sin and cos of 210 degree values

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to sin and cos of 30 degrees.

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So this point in the first quadrant

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has an x-coordinate of cos 30 degrees
and a y-coordinate of sin 30 degrees.

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So relating these two y-coordinates,
we can see from the symmetry here

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that sin of 210 degrees is equal
to negative sin of 30 degrees.

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We've established
from the unit circle

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that sin 210 degrees is equal
to negative sin 30 degrees.

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And sin 30 degrees is an exact value

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that can be calculated from
the following "special triangle",

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that is a "special triangle"

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that is an equilateral triangle
of side length of 2.

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Using trigonometry in this
right angled "special triangle",

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we can see that sin of 30 degrees is
equal to opposite of a hypotenuse

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which is 1 over 2.

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So sin of 210 degrees is equal
to negative sin of 30 degrees

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which is equal to negative one-half.

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Part b.

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Find the exact value
of cos of 315 degrees.

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So we first establish
where in the unit circle

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an angle of 315 degrees lies,

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and we see that it's in
the third quadrant.

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So this point on the unit circle

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has an x-coordinate
of cos of 315 degrees

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and a y-coordinate
of sin of 315 degrees.

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We note that this line is 45 degrees
away from the x-axis

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so we should be able to relate the
values of sin and cos of 315 degrees

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to sin and cos of 45 degrees.

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This point in the first quadrant

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has coordinates
(cos 45 degrees, sin 45 degrees).

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And so we can see from symmetry

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that cos of 315 degrees must be equal
to negative cos of 45 degrees.

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So from the unit circle,
we've established that

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cos of 315 degrees
equals negative cos of 45 degrees.

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And cos of 45 degrees
is an exact value

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that can be calculated from
the following "special triangle".

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And this is a right angled
isosceles triangle

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with equal sides of length 1.

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Using trigonometry in this
right angled "special triangle",

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we can see that cos of 45 degrees
is equal to adjacent over hypotenuse,

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which is one over square root of 2.

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So cos of 315 degrees equals
negative cos of 45 degrees

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which equals negative 1 over root 2.

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Part c.

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Find the exact value
of tan of 150 degrees.

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So we first establish

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where in the unit circle
an angle of 150 degrees lies,

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and we see it lies
in the second quadrant.

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Now, tan is a little different
than sin and cos.

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And tan is the y-coordinate
of the point

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where this line meets with the
tangent at the point (1, 0).

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So this point down here
has an x-coordinate of 1

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and a y-coordinate
of tan 150 degrees.

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Now, we note that this line is
30 degrees away from the x axis

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so we should be able to relate tan of
150 degrees to tan of 30 degrees.

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And we know that this point where the
30 degree line meets with the tangent

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has coordinates (1, tan of 30).

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So again using symmetry,
we can see that tan of 150 degrees

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is equal to negative
tan of 30 degrees.

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So from the unit circle, we've
established this relationship.

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And tan of 30 degrees is an exact
value that can be calculated

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from the following "special triangle"
seen previously.

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And using trigonometry in this
right angled "special triangle",

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we can see that tan of 30 degrees
is opposite over adjacent,

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which is equal to
1 divided by square root of 3.

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So tan of 150 degrees equals
negative tan of 30 degrees

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which equals negative 1 over root 3.