The terminal side of angle 𝐴𝑂𝐵
in standard position intersects with the unit circle at the point 𝐵 with
coordinates negative 𝑥, negative 𝑥. Find the sine of 𝜃.
While we could answer this question
with just the given information, I think it’s more helpful to sketch a picture. We’ll start with the circle that
can be our unit circle. We sketch in the 𝑥-axis and the
𝑦-axis. We’re told that this angle is in
the standard position. And that means that its vertex is
at the origin and that its initial side is along the positive 𝑥-axis.
We also know that point 𝐵 falls
along the unit circle at point negative 𝑥, negative 𝑥. The terminal side of our angle
intersects this point 𝐵. And that means we’re interested in
this distance. To find the sine of this angle,
we’ll need to remember some facts about the unit circle. We need to remember that 90 degrees
in radians is equal to 𝜋 over two.
The first quarter of our turn is an
angle of half 𝜋. The second quarter is another half
𝜋. One-half plus one-half equals an
angle measure of 𝜋. Now, point negative 𝑥, negative 𝑥
is halfway between our next quarter turn. It is one-half of one-half. We’re turning 𝜋 over four,
one-fourth 𝜋. And remember that our 𝜃, our angle
measure, starts at line 𝑂𝐴 and goes to the terminal side 𝑂𝐵.
It will be equal to 𝜋 plus 𝜋 over
four. We can write one 𝜋 as four over
four 𝜋, and then we can add one-fourth 𝜋 to that value. Our angle measure, our 𝜃, is equal
to five 𝜋 over four. Remember, we’re working in
radians. We memorize that unit circle in
radians. We can take this information, five
𝜋 over four, and plug that in to a sine function, sine of five 𝜋 over four.
At this step you’ll need a computer
or a calculator or a chart that lists out all of your sine functions, or perhaps
you’ve memorized all the sine functions in the unit circle. That’s not so hard, and it can be
very useful! All of those methods should return
negative one over the square root of two. Sine of five 𝜋 over four is
negative one over the square root of two.