NARRATOR: Exercise 3.
Find the exact value of: a, sin 210 degrees,
b, cos of 315 degrees,
and c, tan of 150 degrees.
Part a.
Find the exact value of sin of 210 degrees.
So we first establish where in the unit circle 210 degrees lies,
and it lies in the third quadrant.
So the coordinates of this point
where this line of 210 degrees meets the unit circle,
this point has coordinates (cos 210 degrees, sin 210 degrees).
And we note that this line is 30 degrees away from the x-axis
so we should be able to relate these sin and cos of 210 degree values
to sin and cos of 30 degrees.
So this point in the first quadrant
has an x-coordinate of cos 30 degrees and a y-coordinate of sin 30 degrees.
So relating these two y-coordinates, we can see from the symmetry here
that sin of 210 degrees is equal to negative sin of 30 degrees.
We've established from the unit circle
that sin 210 degrees is equal to negative sin 30 degrees.
And sin 30 degrees is an exact value
that can be calculated from the following "special triangle",
that is a "special triangle"
that is an equilateral triangle of side length of 2.
Using trigonometry in this right angled "special triangle",
we can see that sin of 30 degrees is equal to opposite of a hypotenuse
which is 1 over 2.
So sin of 210 degrees is equal to negative sin of 30 degrees
which is equal to negative one-half.
Part b.
Find the exact value of cos of 315 degrees.
So we first establish where in the unit circle
an angle of 315 degrees lies,
and we see that it's in the third quadrant.
So this point on the unit circle
has an x-coordinate of cos of 315 degrees
and a y-coordinate of sin of 315 degrees.
We note that this line is 45 degrees away from the x-axis
so we should be able to relate the values of sin and cos of 315 degrees
to sin and cos of 45 degrees.
This point in the first quadrant
has coordinates (cos 45 degrees, sin 45 degrees).
And so we can see from symmetry
that cos of 315 degrees must be equal to negative cos of 45 degrees.
So from the unit circle, we've established that
cos of 315 degrees equals negative cos of 45 degrees.
And cos of 45 degrees is an exact value
that can be calculated from the following "special triangle".
And this is a right angled isosceles triangle
with equal sides of length 1.
Using trigonometry in this right angled "special triangle",
we can see that cos of 45 degrees is equal to adjacent over hypotenuse,
which is one over square root of 2.
So cos of 315 degrees equals negative cos of 45 degrees
which equals negative 1 over root 2.
Part c.
Find the exact value of tan of 150 degrees.
So we first establish
where in the unit circle an angle of 150 degrees lies,
and we see it lies in the second quadrant.
Now, tan is a little different than sin and cos.
And tan is the y-coordinate of the point
where this line meets with the tangent at the point (1, 0).
So this point down here has an x-coordinate of 1
and a y-coordinate of tan 150 degrees.
Now, we note that this line is 30 degrees away from the x axis
so we should be able to relate tan of 150 degrees to tan of 30 degrees.
And we know that this point where the 30 degree line meets with the tangent
has coordinates (1, tan of 30).
So again using symmetry, we can see that tan of 150 degrees
is equal to negative tan of 30 degrees.
So from the unit circle, we've established this relationship.
And tan of 30 degrees is an exact value that can be calculated
from the following "special triangle" seen previously.
And using trigonometry in this right angled "special triangle",
we can see that tan of 30 degrees is opposite over adjacent,
which is equal to 1 divided by square root of 3.
So tan of 150 degrees equals negative tan of 30 degrees
which equals negative 1 over root 3.