One of the problems with most trig classes is that they tend to concentrate on right triangle trig and do everything in terms of degrees.  Then you get to a calculus course where almost everything is done in radians and the unit circle is a very useful tool.

So first off let’s look at the following table to relate degrees and radians.

Degree	0	30	45	60	90	180	270	360
Radians	0

Know this table!  There are, of course, many other angles in radians that we’ll see during this class, but most will relate back to these few angles.  So, if you can deal with these angles you will be able to deal with most of the others.

Be forewarned, everything in most calculus classes will be done in radians!

Now, let’s look at the unit circle.  Below is the unit circle with just the first quadrant filled in.  The way the unit circle works is to draw a line from the center of the circle outwards corresponding to a given angle.  Then look at the coordinates of the point where the line and the circle intersect.  The first coordinate is the cosine of that angle and the second coordinate is the sine of that angle.  There are a couple of basic angles that are commonly used.  These are  and are shown below along with the coordinates of the intersections.  So, from the unit circle below we can see that  and . 

Remember how the signs of angles work.  If you rotate in a counter clockwise direction the angle is positive and if you rotate in a clockwise direction the angle is negative.

Recall as well that one complete revolution is , so the positive x-axis can correspond to either an angle of 0 or  (or , or , or , or , etc. depending on the direction of rotation).  Likewise, the angle  (to pick an angle completely at random) can also be any of the following angles:

 (start at  then rotate once around counter clockwise)

 (start at  then rotate around twice counter clockwise)

 (start at  then rotate once around clockwise)

 (start at  then rotate around twice clockwise)

etc.

In fact  can be any of the following angles  In this case n is the number of complete revolutions you make around the unit circle starting at .  Positive values of n correspond to counter clockwise rotations and negative values of n correspond to clockwise rotations.

So, why did I only put in the first quadrant?  The answer is simple.  If you know the first quadrant then you can get all the other quadrants from the first.  You’ll see this in the following examples.

Find the exact value of each of the following.  In other words, don’t use a calculator.

1.  and  

The first evaluation here uses the angle .  Notice that .  So  is found by rotating up  from the negative x-axis.  This means that the line for  will be a mirror image of the line for  only in the second quadrant.  The coordinates for  will be the coordinates for  except the x coordinate will be negative.

 

Likewise for  we can notice that , so this angle can be found by rotating down  from the negative x-axis.  This means that the line for  will be a mirror image of the line for  only in the third quadrant and the coordinates will be the same as the coordinates for  except both will be negative.

 

Both of these angles along with their coordinates are shown on the following unit circle.

From this unit circle we can see that  and .

 

This leads to a nice fact about the sine function.  The sine function is called an odd function and so for ANY angle we have

 

2.  and  

For this example notice that  so this means we would rotate down  from the negative x-axis to get to this angle. Also  so this means we would rotate up  from the negative x-axis to get to this angle.  These are both shown on the following unit circle along with appropriate coordinates for the intersection points.