Name:____________________________________________
Per:_____
Date:__________
P3 CW #4: Building the Unit Circle
The unit circle will be your new best friend in Calculus. It helps us to quickly recall the values of trig functions.
So it is really important that you understand how the unit circle is built. It helps to think about the unit circle
as two different circles. One that was cut into 8 equal sections and one that was cut into 12 equal sections.
8 sections
12 sections
Begin by filling in the angle measurements on your unit circle template.
Remember that angle rotations start at the positive x-axis and rotate counter-clockwise.
1. How many degrees are in a full rotation?
2. How many degrees are in a half rotation?
3. Think about dividing your circle into 8 equal sections (pictured above). How many degrees are in an
eighth of a rotation?
4. Using the information above fill in the angle measurements for these 8 points along your unit circle
template. Remember you should be labeling from 0° to 360°.
5. Now, think about dividing your circle into 12 equal sections (pictured above). How many degrees are in
a twelfth of a rotation?
6. Use this information to fill in the remaining angle measurements at the last 8 points. Your unit circle
should now have all the degree angle measurements from 0° to 360°.
7. Now, we need to fill in the radian degree measurements. How many radians are in a full rotation?
8. How many radians are in a half rotation? What about half of that, or a quarter of a rotation?
9. What are the radian measures that correspond to the following degree measures?
a. 30°
b. 45°
c. 60°
𝜋
10. Label all the angle measures on your unit circle template in radians. First, count by 6 , i.e.
𝜋
𝜋
1𝜋 2𝜋 3𝜋
6
,
6
,
6
…,
then by 4 , then by 3 .
11. Next, create your special triangles.
 Cut out the two special triangles at the bottom of the unit circle template.
 You have worked with these triangles before. Decide which triangle is a 30° − 60° − 90° right
triangle and which is a 45° − 45° − 90° right triangle.
 Label the hypotenuse of both triangles as 1.
 Find the lengths of all of the sides and leave them in exact form (no decimals!)
 Label the triangles (on both sides) with the angle measures and the lengths of all the sides.
Finally, we will label the coordinates of the points that correspond to the intersections of these angles and the
unit circle.
12. Using the unit circle template and the triangles you just
labeled, write the ordered pairs as exact values for each point
on the circumference of the circle. Do this by placing one of
your triangles in the appropriate position. For example the
shaded triangle is placed 60° from the initial side (𝛼 = 0°) and
1 √3
).
2
the coordinates (𝑥, 𝑦) should be (2 ,
When you are
finished, double check your coordinates with your team.
13. There are many patterns in the labeled circle you just
completed. Work together with your team to find all of the
patterns and symmetry that you can. Write down your ideas
and be prepared to share them with the class.
Extensions

Unlike right triangle trigonometry, sine, cosine and tangent
may be negative. Find the quadrants where:
a. Sine is positive.
b. Cosine is positive.
c. Tangent is positive.
d. A mnemonic that can be used to describe this result is
“All Students Take Calculus.” How does this mnemonic
work?
Look at the coordinates of all the special angles with denominator 6 on a
𝜋
Unit Circle. What do you notice? The angle 6 is called the reference angle for
each of these angles. A reference angle corresponds to the acute angle that
is formed between the terminal side and the x-axis.
What is the reference angle for
4𝜋
3
?
 In terms of x and y, what is the equation of the unit circle?
Now write the equation of the circle in terms of sin 𝜃 and cos 𝜃? This equation
is referred to as the Fundamental Pythagorean Identity.
Activity adapted from Pre-Calculus with Trigonometry Version 4.0 copyright 2009 CPM Educational Program.