How will we find sine and cosine of these angles? Asking politely doesn't work, looking up the answers online is a no-go (…Shmoop doesn't count), and math problems are notoriously resistant to torture.
We already know some values for angles in the first quadrant, and we'll use that information to figure out other angles. The missing piece to this puzzle is the corresponding acute angle (sometimes called the reference angle).
They're the angles in green. They cling to the x-axis like a joey in its mother's pouch (adorable).
The important thing to note is that the sine and cosine of any angle are equal to the corresponding acute angle's—except for their signs. x and y change sign according to which quadrant they're in.
Sample Problem
What are the sine and cosine of 
?
No, its sine is not Aquarius, and it cannot cosine a loan for you either. Those homophones are a stretch, even for us.
Our first order of business is to find the corresponding acute angle. 
is a little less than π, so we'll use that as our guide.
Looking at the graph makes it obvious that the closest part of the x-axis has angle π. If we subtract our angle from the x-axis, we can find the difference between them:
Our corresponding angle is 
. From the last section, we know that both x and y are 
. So hey, problem over, right?
Wrong, wrong, wrong. Remember, we want the coordinates for 
, which is in the second quadrant of the graph. The x-coordinate needs to be negative.
That means x is 
(which means so is cosine), and y (and also sine) is 
.
Now we're done. With this problem, at least. We have another lined up. Sorry.
Sample Problem
What are the sine and cosine of 
?
Yeah, we just switched around the 3 and the 4 from the last problem. It's not lazy, it's efficient.
Anyway, let's make another graph. Our angle is a little larger than π, but less than 
.
Now, some people would look at this and say, "The angle is closest to , so the corresponding acute angle must be 
."
Those people would be wrong (as Dr. Cox never tires of telling us). The corresponding acute angle goes with the x-axis, not the y-axis; remember the cute kangaroo joey? It was adorable, and with the x-axis. We find the correct angle by subtracting π out from 
:
We're in the third quadrant, so both x and y are negative. That means 
and 
.
You've got to keep an eye on those negative signs. They're easy to lose, but they always cause trouble if they're forgotten somewhere.
One last thing to note. Angles can get big; really big. How would we find the corresponding acute angle for a ridiculous angle like 57π?
We'll answer that question with another one: 2π, or not 2π? (That is the question.) And the answer is always "2π." That's a full circle, so subtracting 2π from an angle doesn't change its position on the unit circle.
57π – 2π = 55π
55π – 2π = 53π
Just keep on going, until we hit:
3π – 2π = π
So 57π is in the same position on the unit circle as plain ol' π. We know sine and cosine of π, and they'll be the same as sine and cosine of 57π.
That's how we work with big angles: whittle them down to size until they are manageable. Whittle whittle whittle.
We just like saying "whittle" out loud.
Whittle.