– 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. Do we
remember what this is called?
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. Do we
remember what this is called? Standard Position
S
o
l
u
t
i
o
n
s
1. 90°
2. 45°
3. 30°
4. 150°
5. 135°
6. 120°
1.
2.
3.
4.
5.
6.
The Unit Circle
The Unit Circle
- Radius is always one unit
- Center is always at the
origin
Let’s pick a point on
the unit circle. The
positive angle
always goes
counter-clockwise
from the x-axis.
1
 cos 30,sin 30
30
-1
In order to determine the sine and
cosine we need a right triangle.
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.
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°.
The angle formed by the terminal side of the angle in standard position
and the closest x axis is called the reference angle.
Here the reference angle is 30º
Angles and the Unit Circle
Sketch each angle in standard position and find the reference angle for each.
a. 48°
Reference is the same
b. 310°
Reference is 50º
c. –170°
Reference is 10º
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
2
1
60
30
P
X- coordinate
30
-1
1
Y- coordinate
-1
1
The Unit Circle
You can see why the x
co-orodinate is cosine
and the y co-ordinate is
sine when we overlap
the two triangles to
create similar triangles.
1
2
P
1
30
-1
60
1
2
The smaller triangle
has a hypotenuse of 1
1 unit, the radius of the
unit circle which is half
our identity triangle.
3
The X- coordinate is the horizontal
distance of the smaller triangle
-1
The Y- coordinate is the vertical
distance of the smaller triangle
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
120°
( 1 , 3 )
2
π/3
60°
2
1
3
)
2
( 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, 1 )
2
(
2
150°
5π/6
3
,
2
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