Unit Circle

1
The Unit Circle
--------1-------
(0 , 0)
2
Vocabulary

• The Unit Circle
• A circle centered at the origin with a
  radius of
• exactly one unit.
• Equation for unit circle
• x2 y2 1
• Reference Angle
• the acute angle made between the Terminal Side
  and the x-axis
• Quantrantal Angle
• has its terminal side on one of the coordinate
  axes.

3
The Unit Circle

• Start by finding the points on the unit circle
  at each of the quantrantal angles.
• Starting position
• (1,0)
• 90
• (0,1)
• 180
• (-1,0)
• 270
• (0,-1)

(0,1)
(1,0)
(-1,0)
(0,-1)
4
The Unit Circle

• Draw an angle of 30 degrees from the starting
  position.
• Create a right triangle by dropping a
  perpendicular line to the x axis.
• Find the lengths of the two legs using what you
  know about 30-60-90 triangles

(0,1)
1
½
30
(1,0)
(-1,0)
60
2a
(0,-1)
1a
30
5
The Unit Circle

• What is the point on the unit circle for the
  reference angle 30 ?
• From the origin we travelled units right.
  And then we travelled ½ units up.
• The point will be

(0,1)
1
½
30
(1,0)
(-1,0)
(0,-1)
6
The Unit Circle
What is the point on the unit circle for the
reference angles of 30 ? Draw the other
reference angles of 30 and create right
triangles for them
(0,1)
1
½
30
30
(-1,0)
(1,0)
30
30
½
½
1
(0, -1)
7
The Unit Circle
What is the point on the unit circle for the
reference angles of 45 ? Draw the reference
angles of 45 and create right triangles for
them Find the points on the unit circle using
what you know about 45-45-90 triangles
(0,1)
1
45
45
(-1,0)
(1,0)
45
45
(0, -1)
8
The Unit Circle
What is the point on the unit circle for the
reference angles of 60 ? Draw the reference
angles of 60 and create right triangles for
them Find the points on the unit circle using
what you know about 30-60-90 triangles
(0,1)
1
1
60
60
(-1,0)
(1,0)
60
60
1
1
(0, -1)
9
The Unit Circle
Fill in the degrees and radians on the inside of
the circle. Find cos(60) ½ Find
sin(60) .8660254038 Change to a
decimal .8660254038 Find cos(90)? 0 Find
sin(90)? 1 Does this pattern continue for all the
points?
10
The Unit Circle
Each of the points on the unit circle are (cos
?, sin ?) Fill in the degrees and radians in
your unit circle.
11
Angles and the Unit Circle
Find the cosine and sine of 135.
First, find the reference angle 45 From the unit
circle, the x-coordinate
of point of the reference angle 45 is  so cos
135 Sin 135 is