Key Values of the Unit Circle

45° = π/4 angle

First let's calculate the coordinates associated with 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 right triangle. Note that in the triangle the angle opposite of our 45° angle is also 45° 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 (call the side a). From the Pythagorean Theorem we know that a² + a² = 1² Thus 2a² = 1 This leads to a = 1/√2 = √2/2.

Thus we know

Use what you just learned above and what you already know about the symmetry of the unit circle to label all of the marked points on the following unit circle with their coordinates.

After reading the 30° section below label all of the marked points on the following unit circle with their coordinates.

30° = π/6 angle

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 right triangle. Note that the opposite 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 original 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 P(π/6) = (√3/2, ½)

60° = π/3 angle

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 out the coordinatesat 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. You can hit refresh to see the animation in the picture. From the figure you can see that P(π/3) = (½, √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°].

I. Using the circles you just made for reference, find the following points exactly:

P(5π/6)
P(5π/3)
P(315°)
P(210°)

II. Now use your knowledge that if you add or subtract a multiple of 360° or 2π^R you end up at the same point on the circle to find the following values exactly:

P(-135°)
P(7π/3)
P(-7π/6)
P(990°)

Memory Gimick

Now that you see how the key values of the unit circle are derived, you may be wondering how you are going to remember them. You may want to practice labeling a new unit circle as you read through this part.

First of all realize that you already know the values of the coordinates on the x and y axis since you know that we are dealing with a unit circle, i.e. a circle centered at the origin with radius one.
Next label all of the key points in the first quadrant like this: leaving the numerators blank.

Now realize that

and use the fact that the x-coordinates represent the horizontal lengths and the y-coordinates represent the vertical lengths. Put the longer labels on the longer lengths and shorter labels on the shorter lengths for each coordinate.

Lastly use your knowledge of point plotting and unit circle symmetry to fill in the rest of your labels. You should get the same labeled circle that we got before.

Get good at labeling the unit circle on the fly even with a hand-drawn circle. You will find this skill to be valuable throughout the semester. After practicing labeling a unit circle, you should also practice counting in radians on the unit circle. See my interactive flash movie to help you with this.

Answers:
I. 1. (-√3/2, ½) 2. (½, -√3/2) 3. (√2/2, √2/2) 4. (-√3/2, -½)
II. 5. (-√2/2, -√2/2) 6. (½, √3/2) 7. (-√3/2, ½) 8. (0, -1)