13.2 – Angles and the Unit Circle
Angles and the Unit Circle
For each measure, draw an angle with its vertex at the origin of the
coordinate plane. Use the positive x-axis as one ray of the angle.
1. 90°
2. 45°
3. 30°
4. 150°
5. 135°
6. 120°
Angles and the Unit Circle
For each measure, draw an angle with its vertex at the origin of the
coordinate plane. Use the positive x-axis as one ray of the angle.
1. 90°
2. 45°
3. 30°
4. 150°
5. 135°
6. 120°
1.
2.
3.
4.
5.
6.
Solutions
The Unit Circle
The Unit Circle
- Radius is always one
unit
- Center is always at
the origin
- Points on the unit
circle relate to the
periodic function
Let’s pick a point on
the unit circle. The
positive angle
always goes
counter-clockwise
from the x-axis.
1
30
-1
1
-1
The x-coordinate of
this has a value of the
cosine of the angle.
The y-coordinate has a
value of the sine of
the angle.
In order to determine the sine and
cosine we need a right triangle.
The Unit Circle
The angle can also be
negative. If the angle is
negative, it is drawn
clockwise from the x
axis.
1
-1
- 45
-1
1
Angles and the Unit Circle
Find the measure of the angle.
The angle measures 60° more than a right angle of 90°.
Since 90 + 60 = 150, the measure of the angle is 150°.
Angles and the Unit Circle
Sketch each angle in standard position.
a. 48°
b. 310°
c. –170°
Let’s Try Some
Draw each angle of the unit circle.
a. 45o
b. -280 o
c. -560 o
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)
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
The Unit Circle
Let’s look at an example
The x-coordinate of
this has a value of the
cosine of the angle.
The y-coordinate has a
value of the sine of
the angle.
1
In order to determine the sine and
cosine we need a right triangle.
30
-1
1
-1
The Unit Circle
1
30
-1
1
Create a right triangle, using
the following rules:
1. The radius of the circle is
the hypotenuse.
2. One leg of the triangle
MUST be on the x axis.
3. The second leg is parallel
to the y axis.
Remember the ratios of a 30-60-90
triangle2
-1
30
60
1
The Unit Circle
1
2
60
30
P
X- coordinate
30
-1
1
Y- coordinate
-1
1
Angles and the Unit Circle
Find the cosine and sine of 135°.
From the figure, the x-coordinate of point A
is –
2 , so cos 135° = –
2
2 , or about –0.71.
2
Use a 45°-45°-90° triangle to find sin 135°.
opposite leg = adjacent leg
=
2
2
0.71
Substitute.
Simplify.
The coordinates of the point at which the terminal side of a 135° angle intersects
are about (–0.71, 0.71), so cos 13
–0.71 and sin 135°
0.71.
Angles and the Unit Circle
Find the exact values of cos (–150°) and sin (–150°).
Step 1: Sketch an angle of –150° in
standard position. Sketch a
unit circle.
x-coordinate = cos (–150°)
y-coordinate = sin (–150°)
Step 2: Sketch a right triangle. Place the
hypotenuse on the terminal side
of the angle. Place one leg on the
x-axis. (The other leg will be
parallel to the y-axis.)
Angles and the Unit Circle
(continued)
The triangle contains angles of 30°, 60°, and 90°.
Step 3: Find the length of each side of the triangle.
hypotenuse = 1
1
2
shorter leg =
longer leg =
The hypotenuse is a radius of the unit circle.
1
2
The shorter leg is half the hypotenuse.
3=
3
2
The longer leg is
3 times the shorter leg.
Since the point lies in Quadrant III, both coordinates are negative. The longer leg
lies along the x-axis, so
cos (–150°) = –
3
2
, and sin (–150°) = – 1 .
2
Let’s Try Some
Draw each Unit Circle. Then find the cosine and sine of each angle.
a. 45o
b. 120o
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
2
3π/4
135°
2
)
2
(
,
2
45°
π/4
2
2
2
2
5π/4
225°
( 2 ,  2 )
2
2
7π/4
315°
( 2 ,
2

2)
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 - Coordinates
Use the side lengths we labeled on the QI triangle to determine coordinates.
2π/3
π/3
60°
120°
1
(  , 3 )
2 2
1
( 2
3
)
2
,
3
2
1
2
5π/3
4π/3
( 1 ,  3)
2
2
240°
300°
( 1
2
,

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
about in radians?
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
(
,
2
1
(  3,
)
2 2
150°
5π/6
1
)
2
30° π/6
1
2
3
2
7π/6
210°
( 3 ,  1 )
2
2
330°
11π/6
( 3 ,1 )
2
2
Final Product
The Unit Circle