Video Transcript
Find the value of the sin of 11𝜋
by six.
In this question, we’re asked to
evaluate the sin of on angle in radians. We can notice that 11𝜋 by six is
not an acute angle, so we can’t do this just by using right-triangle
trigonometry. So instead, we need to recall that
we can evaluate trigonometric functions by sketching the angle in standard position
and then using the intersection with the unit circle. To do this, we need to find the
coordinates of the point of intersection between the terminal side of our angle in
standard position and the unit circle. So we’ll start by sketching our
angle in standard position.
Remember, this is measured
counterclockwise from the positive 𝑥-axis. To determine which side the
terminal side lies in, it can help to add the right angles to our diagram, where we
note that one full revolution gives us two 𝜋. If we were to evaluate 11 over six,
we would see it’s equal to 1.83 recurring. So it’s between three over two and
two. Multiplying this inequality through
by 𝜋, we see that our angle must lie in the fourth quadrant. This gives us the following
sketch.
Remember, the coordinates of the
point of intersection between the terminal side and the unit circle will help us
evaluate the trigonometric function at this angle, in this case, the coordinates of
the point of intersection are the cos of 11𝜋 by six and the sin of 11𝜋 by six. This means we’re asked to determine
the 𝑦-coordinate of this point of intersection. To do this, we’ll drop a
perpendicular line from the point of intersection to our 𝑥-axis. Then the length of this line can be
used to find the value of the sin of 11𝜋 by six.
To help us determine this value,
let’s isolate the following right triangle from our diagram. We want to determine the height of
this right triangle. And to do this, we can notice we
can find the hypotenuse of our triangle. We can see it’s a radius of the
circle. And this is the unit circle, so all
of the radii have length one. Next, we know the height of our
triangle was given by sin of 11𝜋 by six. But we do need to be careful
because this 𝑦-coordinate is negative. So instead, we’ll write this as the
absolute value of the sin of 11𝜋 by six to make this a positive value.
Finally, we can determine one of
the angles in our right triangle. We can see that, together with 11𝜋
by six, this angle makes a full revolution, which means this angle has measure two
𝜋 minus 11𝜋 by six. And if we evaluate this, we get 𝜋
over six. It’s worth noting that this angle
has a name. It’s called the reference angle of
11𝜋 by six. It’s the acute angle that the
terminal side makes with the 𝑥-axis. We now have an angle and one side
in our right triangle, and we want to determine the other side in our right
triangle. So we can do this by using
trigonometry.
Remember, the sine of an angle in a
right triangle is the ratio of the length of the opposite side and the hypotenuse
side in our right triangle. In our case, this tells us the sin
of 𝜋 by six is equal to the absolute value of the sin of 11𝜋 by six all divided by
one. And we can evaluate this
expression. First, dividing by one doesn’t
change the value. Next, 𝜋 by six is a special
angle. We know the sin of 𝜋 by six is
equal to one-half. In fact, we can prove this by using
equilateral triangles. Therefore, we’ve shown that the
height of our right triangle is equal to one-half.
But we’re not done yet. Remember, we’re asked to evaluate
the sin of 11𝜋 by six, which is the 𝑦-coordinate of the point of intersection
between the terminal side and the unit circle. This is why it’s important we
remember our absolute value sign. We can see in the diagram that this
point of intersection has a negative 𝑦-coordinate. This means we need to take the
negative value of our answer, giving us that the sin of 11𝜋 by six is equal to
negative one-half, which is our final answer. Therefore, by sketching the angle
11𝜋 by six in standard position and then finding its reference angle, which was 𝜋
by six, and then applying trigonometry, we were able to show the sin of 11𝜋 by six
is equal to negative one-half.
