Our textbook shows how to find trig values of 30°, 60°, 90° triangles and of 45°, 45°, 90° triangles. I had you start the course in the middle of chapter 5 because I believe that if you begin learning trigonometry from the circle point of view, you will learn it better with a lot less memorizing since you will have the tools to think things through with the help of the unit circle. Here, I'll give a brief rehash of the derivations of these trig values and then I let you use them to get the trig values of any multiple of these angles.

First let's calculate the trig values of a 45° = π/4^R angle. Measure π/4 units up from the positive x-axis on the unit circle and drop a line segment down to the positive x-axis from that point forming a triangle. Note that in the blue triangle the other angle is 45° also, since the sum of the angles of a triangle add to 180°. Two of the angles are the same so the sides opposite of those angles must also be the same. From the Pythagorean Theorem we know that a² + a² = 1² Thus 2a² = 1 This leads to a = 1/√2 = √2/2. Thus we know cos (π/4) = sin (π/4) = 1/√2.

Now let's calculate the trig values at a 30° = π/6^R angle. Measure π/6 units up from the positive x-axis along the circular arc and drop a line segment down from that point as above to form the red triangle. Note that the other angle in the triangle is a 60° angle. Now if we make a copy of the triangle and flip it over the x-axis we have a combined larger triangle where all the angles are 60°, hence all the sides are the same with length = 1. The side opposite of our 30° angle in the red triangle must be of length ½, since it is half the length of a side on the larger triangle. Now we can easily calculate the third side using the Pythagorean Theorem again. . Now you can see that cos (π/6) = √3/2 and sin (π/6) = ½

When we looked at the 30° angle above, we ended up drawing a triangle that also had a 60° angle in it. So now since we want to figure the trig functions at a 60° angle, we can just use the same triangle as above, but flip it around so that the 60° = π/3^R angle is the one measured from the positive x-axis. From the figure you can see that cos (π/3) = ½ and sin (π/3) = √3/2 since we just switched the legs on the triangle.

You will need to print up some unit circles to write on to do the next part. Click here if you need them.

Now you should be able to use your knowledge of point plotting from your past algebra experiences to name all the points that are marked on this unit circle. Click here to see a labeled circle after you have tried to label one on your own. You should also be able to label these points with their corresponding angles in both radians and degrees. Just use your knowledge that there are 360°= 2π^R in a circle and all these labeled points are easy fractions of 360° or 2π^R, or you could use the angles we just derived in the first quadrant and think of all the other marked angles as multiples of those.

Click here to see the unit circle labeled with its angles in radians from [0, 2π].

Click here to see the unit circle labeled with its angles in degrees from [0, 360°]. Note the degree symbols are missing in this picture. Technically, they should be there because anytime the degree symbol is not there the angle is assumed to be in radians. I will correct this image some day when I get a chance, but it might not happen this semester.

You should be able to do these problems without a calculator.
Using the circles you just made for reference, find the following values exactly:

1. cos (5π/6)
2. cot (5π/3)
3. sec 315°
4. sin 210°

5. tan (-135°)
6. csc (7π/3)
7. cos (-7π/6)
8. sin 990°