Find the cos of negative 960 degrees without using a calculator.
Our first step
would be to rewrite this expression with the smallest positive angle possible by adding 360
degrees to negative 960 over and over until we get something positive. If we add 360, we get
negative 600, so we need to add another 360 since it’s not positive yet. After adding 360 again,
we’re at negative 240. Finally, after adding 360 again, we get 120.
So to help visualize this, here we
can see 120 degrees. This is equal to negative 960 degrees because if we had started here at
zero degrees and we went left, so counterclockwise, we would be at 120 degrees. However, the negative
960 degrees, one full rotation backward would- we would be at negative 360. After another full rotation, we
would be at negative 720, another 360 degrees in the negative direction.
Now instead of going another
360, if we went another negative 240, we would be at the exact same spot. Now one step further would be to
look at the reference angle, which would be 60 degrees, because the entire top half should be 180
degrees. So if we already have 120, there’d be 60 left.
Knowing that our point 𝑥, 𝑦 is cos 𝜃, sin 𝜃,
so 𝑥 is the cos of 𝜃 and 𝑦 is the sin of 𝜃. The sin and cos of the special angle of
60 degrees is that the sin of 60 degrees is square root three over two, and the cos of 60
degrees is one-half. So we plug in one-half for cos of 𝜃, which is our 𝑥-coordinate, and
square root three over two for the sin of 𝜃, which is our 𝑦-coordinate.
However, we need to
be careful of our signs for 𝑥 and 𝑦. The angle that we’re looking at is in the top left-hand
corner; that’s called quadrant number two. And in quadrant two, 𝑥 is negative and 𝑦 is positive,
so we should go ahead and add the negative sign to our 𝑥 value.
So since it says to find the cos of
negative 960 degrees, which was equal to the 120, and then we looked at our reference angle of 60
degrees, we found that the cosine is equal to negative one-half. Therefore, the cos of negative
960 degrees is equal to negative one-half .