### Discovering the Unit Circle

```The Unit Circle
PreCalculus
Right now…
Get a protractor, scissors, and one copy of each
circle (blue, green, yellow, white).
Sit down and take everything BUT that stuff & your
writing utensils(4 different colors if you have
Cut out the blue, green, and yellow circles.
Put your name on the white paper. DO NOT cut
the white circle!
The Unit Circle
Definition: A circle centered at the origin
with a radius of exactly one unit.
(0, 1)
(-1,0)
|-------1-------|
(0 , 0)
(1,0)
(0, -1)
** Note – You
should be writing
this information
on the white
paper!
Using what you learned in 4.1 about
sketching angles, what are the angle
measurements of each of the four
angles we just found?
90° π/2
0° 0
360° 2π
180°
π
270°
3π/2
Using the Blue Circle
Fold the circle in half twice.
You should now be holding something that
looks like a quarter of a pie.
Hold the piece with the two folds on the
left and the single fold on the bottom.
Using the Blue Circle
MAKE SURE THE DOUBLE FOLD IS ON THE LEFT, THE SINGLE FOLD ON THE
BOTTOM.
With your protractor in the corner of the pie piece, draw a 45° angle.
Start cutting here,
then over to the line
Hold the corner of the pie piece and cut along the line you just drew (cut slightly above
the corner, not through it).
Once you reach the outside of the circle, cut down to the single fold, forming a 45-4590 right triangle.
45-45-90 Blue Triangle
We know that a 45-45-90 triangle has side lengths:
2
1
1
But… Our right triangle has a hypotenuse of 1 (because that’s the radius
of the circle).
So the new side lengths of the 45-45-90 triangle are:
1
2
2
2
2
45-45-90 Blue Triangle
Holding the triangle with the single fold down and double fold to the left, label
each side on the triangle.
Unfold the triangle (so it looks like a butterfly) and glue it to the white circle
with the triangle you just labeled in quadrant I.
45° Reference Angles
We know that the quadrant one angle formed by the triangle is 45°.
That means each other triangle is showing a reference angle of 45°. What
Label the remaining three angles.
135°
3π/4
45°
π/4
225°
5π/4
315° 7π/4
45° Reference Angles - Coordinates
Remember that the unit circle is overlayed on a coordinate plane (that’s how
we got the original coordinates for the 90°, 180°, etc.)
Use the side lengths we labeled on the QI triangle to determine coordinates.
(
2,
2
2
2
3π/4
135°
)
2
2
(
,
2
45°
π/4
2
)
2
2
2
2
5π/4
(
2
2
,

225°
)
2
2
7π/4
315°
( 2 ,

2
2)
2
Using the Green Circle
Fold the circle in half twice.
You should now be holding something that
looks like a quarter of a pie.
Hold the piece with the two folds on the
left and the single fold on the bottom.
Using the Green Circle
MAKE SURE THE DOUBLE FOLD IS ON THE LEFT, THE SINGLE FOLD ON THE
BOTTOM.
With your protractor in the corner of the pie piece, draw a 60° angle.
Start cutting here,
then over to the line
Hold the corner of the pie piece and cut along the line you just drew (cut slightly above
the corner, not through it).
Once you reach the outside of the circle, cut down to the single fold, forming a 30-6090 right triangle with the 60° at the bottom.
30-60-90 Green Triangle
We know that a 30-60-90 triangle has side lengths:
2
3
60°
1
But… Our right triangle has a hypotenuse of 1 (because that’s the radius
of the circle.
1
So the new side lengths of the 30-60-90 triangle are:
3
2
60°
1
2
30-60-90 Green Triangle
Holding the triangle with the single fold down and double fold to the left, label
each side on the triangle.
Unfold the triangle (so it looks like a butterfly) and glue it to the white circle
with the triangle you just labeled in quadrant I, on top of the blue butterfly.
60° Reference Angles
We know that the quadrant one angle formed by the triangle is 60°.
That means each other triangle is showing a reference angle of 60°. What
Label the remaining three angles.
120°
2π/3
60° π/3
5π/3
4π/3
240°
300°
60° Reference Angles - Coordinates
Use the side lengths we labeled on the QI triangle to determine coordinates.
2π/3
(

1
,
3
2
π/3
60°
120°
)
2
(
1
3
,
2
2
)
3
2
1
2
5π/3
4π/3
(

1
2
,

3
2
)
240°
300°
(
1
2
,

3)
2
Using the Yellow Circle
Fold the circle in half twice.
You should now be holding something that
looks like a quarter of a pie.
Hold the piece with the two folds on the
left and the single fold on the bottom.
Using the Yellow Circle
MAKE SURE THE DOUBLE FOLD IS ON THE LEFT, THE SINGLE FOLD ON THE
BOTTOM.
With your protractor in the corner of the pie piece, draw a 30° angle.
Start cutting here,
then over to the line
Hold the corner of the pie piece and cut along the line you just drew (cut slightly above
the corner, not through it).
Once you reach the outside of the circle, cut down to the single fold, forming a 30-6090 right triangle with the 30° at the bottom.
30-60-90 Yellow Triangle
We know that a 30-60-90 triangle has side lengths:
2
30°
1
3
But… Our right triangle has a hypotenuse of 1 (because that’s the radius
of the circle.
So the new side lengths of the 30-60-90 triangle are:
1
1
2
30°
3
2
30-60-90 Yellow Triangle
Holding the triangle with the single fold down and double fold to the left, label
each side on the triangle.
Unfold the triangle (so it looks like a butterfly) and glue it to the white circle
with the triangle you just labeled in quadrant I, on top of the green
butterfly.
30° Reference Angles
We know that the quadrant one angle formed by the triangle is 30°.
That means each other triangle is showing a reference angle of 30°. What
Label the remaining three angles.
150°
π/6
30°
5π/6
210°
7π/6
11π/6
330°
30° Reference Angles - Coordinates
Use the side lengths we labeled on the QI triangle to determine coordinates.
(
3, 1
2
)
3
(
2
2
150°
5π/6
,
1
2
)
30° π/6
1
2
3
2
(
3
2
7π/6
210°
330°
1
(
,

2
)
3
2
11π/6
,
1
2
)
Final Product
The Unit Circle
```