10-3 The Unit Circle
Warm Up
Find the measure of the reference angle
for each given angle.
1. 120°
2. 225°
Convert the following from degrees to
radians or radians to degrees.
3. 150°
Holt McDougal Algebra 2
4. 2π radians
10-3 The Unit Circle
A unit circle is a circle with a radius of 1 unit.
For every point P(x, y) on the unit circle, the value of
r is 1.
2
2
x  y 1
(0,1)
(-1,0)
(1,0)
(0,-1)
Holt McDougal Algebra 2
10-3 The Unit Circle
Why the unit circle?
• We use the unit circle to help us evaluate
any trigonometric function.
• Ex. sin 60° =
• Ex. cos Π/2 =
Holt McDougal Algebra 2
10-3 The Unit Circle
Explore the Unit Circle
• http://www.geogebratube.org/student/m26
809
• Take a few moments to go to this website
and explore the unit circle.
• Write down any interesting things you
notice about the sine, cosine, and tangent
values as you change the angle.
Holt McDougal Algebra 2
10-3 The Unit Circle
Creating/ completing the unit
circle
We will keep in mind 2 things
1. The unit circle is symmetric in many
ways (reference angles)
2. The quadrant helps us to find out which
is positive and which is negative.
Holt McDougal Algebra 2
10-3 The Unit Circle
The diagram shows how
the signs of the
trigonometric functions
depend on the quadrant
containing the terminal
side of θ in standard
position.
Holt McDougal Algebra 2
10-3 The Unit Circle
Holt McDougal Algebra 2
10-3 The Unit Circle
So the
coordinates of P
can be written
as (cosθ, sinθ).
tanθ = sinθ
cosθ
Holt McDougal Algebra 2
10-3 The Unit Circle
Example 2A: Using the Unit Circle to Evaluate
Trigonometric Functions
Use the unit circle to find the exact value of
each trigonometric function.
cos 225°
Holt McDougal Algebra 2
10-3 The Unit Circle
Example 2B: Using the Unit Circle to Evaluate
Trigonometric Functions
Use the unit circle to find the exact value of
each trigonometric function.
tan
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 1a
Use the unit circle to find the exact value of
each trigonometric function.
sin 315°
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 1b
Use the unit circle to find the exact value of
each trigonometric function.
tan 180°
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 1c
Use the unit circle to find the exact value of
each trigonometric function.
Holt McDougal Algebra 2
10-3 The Unit Circle
You can use reference angles and Quadrant I of the
unit circle to determine the values of trigonometric
functions around the unit circle.
Holt McDougal Algebra 2
10-3 The Unit Circle
The diagram shows how
the signs of the
trigonometric functions
depend on the quadrant
containing the terminal
side of θ in standard
position.
Holt McDougal Algebra 2
10-3 The Unit Circle
Example 3: Using Reference Angles to Evaluate
Trigonometric functions
Use a reference angle and or the unit circle to
find the exact value of the sine, cosine, and
tangent of 330°.
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 3a
Use a reference angle and or the unit circle to
find the exact value of the sine, cosine, and
tangent of 270°.
270°
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 3b
Use a reference angle and or the unit circle to
find the exact value of the sine, cosine, and
tangent of each angle.
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 3c
Use a reference angle and or the unit circle to
find the exact value of the sine, cosine, and
tangent of each angle.
–30°
–30°
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 3c
Use a coterminal angle and the unit circle to
find the exact value of the sine, cosine, and
tangent of each angle.
390°
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 3c
Use a coterminal angle and or the unit circle to
find the exact value of the sine, cosine, and
tangent of each angle.
5π/2
Holt McDougal Algebra 2
10-3 The Unit Circle
Arc Length
Holt McDougal Algebra 2
10-3 The Unit Circle
If you know the measure of a central angle of a
circle, you can determine the length s of the arc
intercepted by the angle.
Holt McDougal Algebra 2
10-3 The Unit Circle
Holt McDougal Algebra 2
10-3 The Unit Circle
Example 4: Automobile Application
A tire of a car makes 653 complete rotations
in 1 min. The diameter of the tire is 0.65 m. To
the nearest meter, how far does the car travel
in 1 s?
Holt McDougal Algebra 2
10-3 The Unit Circle
Example 4 Continued
Holt McDougal Algebra 2
10-3 The Unit Circle
Example 4 Continued
Step 3 Find the length of the arc intercepted by
radians.
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 4
An minute hand on Big Ben’s Clock Tower in
London is 14 ft long. To the nearest tenth of a
foot, how far does the tip of the minute hand
travel in 1 minute?
Step 1 Find the radius of the clock.
r =14
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 4 Continued
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 4 Continued
Step 3 Find the length of the arc intercepted by
radians.
Holt McDougal Algebra 2
10-3 The Unit Circle
Lesson Quiz: Part I
Convert each measure from degrees to radians
or from radians to degrees.
1. 100°
2.
3. Use the unit circle to find the exact value of
4. Use a reference angle to find the exact value of
the sine, cosine, and tangent of
Holt McDougal Algebra 2
.
10-3 The Unit Circle
Lesson Quiz: Part II
5. A carpenter is designing a curved piece of
molding for the ceiling of a museum. The curve
will be an arc of a circle with a radius of 3 m.
The central angle will measure 120°. To the
nearest tenth of a meter, what will be the length
of the molding?
Holt McDougal Algebra 2