Reference Angles for Sine and Cosine

Finding reference angles involves a series of steps.

Finding a reference angle of sine or cosine

1. Find where sine or cosine lies on a unit circle
The unit circle is a circle with a radius of 1. The unit circle's center usually lies on the origin at point (0,0).
2. Find if sine(Y) or Cosine(X) is positive or negative at the location where you stopped in step 1.
3. Write down whether it's + or - then write the function.
Ex. - cos + sin or sin.
4. Add or subtract 180 until you have an angle with an absolute value less than 90.
5. Write down that angle's absolute value.

Examples

1. Find a reference angle for:

a) Sin 430
First we imagine a unit circle or draft one up if needed.

Next, we'll go around 430 degrees and stop in that quadrant.

We notice that the number is greater than 360 degrees so we can just subtract 360 from our number and use that angle instead.
430-360=70

So we can go around 70 degrees and stop at Quadrant I.

Sine correlates to your Y coordinates. in quadrant I your Y's are always positive. So we either write + sin or sin as the first part of our answer.

Lastly we'll subtract 180 until we reach a number whose absolute value is less than 90 and we'll write down that absolute value.
430-360=70; |70|=70

And our answer is; sin 70

b) Cos 584028394392384710

Now we have a pretty large number here, right?

The easiest way to do a problem like this, instead of subtracting adding 360 or subtracting 180 over and over is to use division.

The first thing we need it do is find where it lies on the unit circle.

We know every 360 degrees is a full circle so let's take out as many 360s as we can

584028394392384710/360

After you divide take your remainder and that's what you'll go with on your unit circle.
The remainder comes out to 311. What this means is if you subtracted 360 over and over until you reached 311. It's just a shortcut method for big numbers.

311 on your unit circle ends in Q IV. Since we're working with cosine we're working with X's. your X's are positive in the quadrant IV so your answer must be positive. So we'll write that down.
Cos =
or
+Cos =
Now to find the angle. Subtracting 180 twice is the same as subtracting 360 once. Which means we'd come to 311 eventually, as we did earlier.

The easiest way to go from there is subtract 180 until you get an absolute value less than 90.
311-180 = 131
-180= -49

-49's absolute value is less than 90 so we'll use that absolute value.

cos 49

So we know the reference angle of Cos 584028394392384710 degrees is cos 49 degrees.