10-3The
10-3
TheUnit
UnitCircle
Circle
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
10-3 The Unit Circle
Warm Up
Find the measure of the reference angle
for each given angle.
1. 120° 60°
2. 225° 45°
3. –150° 30°
4. 315° 45°
Find the exact value of each
trigonometric function.
5. sin 60°
6. tan 45°
7. cos 45°
8. cos 60°
Holt McDougal Algebra 2
1
10-3 The Unit Circle
Objectives
Convert angle measures between
degrees and radians.
Find the values of trigonometric
functions on the unit circle.
Holt McDougal Algebra 2
10-3 The Unit Circle
Vocabulary
radian
unit circle
Holt McDougal Algebra 2
10-3 The Unit Circle
So far, you have measured angles in
degrees. You can also measure angles
in radians.
A radian is a unit of angle measure based on arc
length. Recall from geometry that an arc is an
unbroken part of a circle. If a central angle θ in a
circle of radius r, then the measure of θ is defined
as 1 radian.
Holt McDougal Algebra 2
10-3 The Unit Circle
The circumference of a
circle of radius r is 2r.
Therefore, an angle
representing one complete
clockwise rotation
measures 2 radians. You
can use the fact that 2
radians is equivalent to
360° to convert between
radians and degrees.
Holt McDougal Algebra 2
10-3 The Unit Circle
Holt McDougal Algebra 2
10-3 The Unit Circle
Example 1: Converting Between Degrees and
Radians
Convert each measure from degrees to
radians or from radians to degrees.
A. – 60°
.
B.
Holt McDougal Algebra 2
10-3 The Unit Circle
Reading Math
Angles measured in radians are often not labeled
with the unit. If an angle measure does not have
a degree symbol, you can usually assume that
the angle is measured in radians.
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 1
Convert each measure from degrees to
radians or from radians to degrees.
a. 80°
4
9
.
b.
20
.
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 1
Convert each measure from degrees to
radians or from radians to degrees.
c. –36°
5
.
d. 4 radians
.
Holt McDougal Algebra 2
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. Therefore, for an
angle θ in the standard
position:
Holt McDougal Algebra 2
10-3 The Unit Circle
So the coordinates of
P can be written as
(cosθ, sinθ).
The diagram shows
the equivalent
degree and radian
measure of special
angles, as well as
the corresponding xand y-coordinates of
points on the unit
circle.
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°
The angle passes through the point
on the unit circle.
cos 225° = x
Holt McDougal Algebra 2
Use cos θ = x.
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
The angle passes through the point
on the unit circle.
Use tan θ =
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°
The angle passes through the point
on the unit circle.
sin 315° = y
Holt McDougal Algebra 2
Use sin θ = y.
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°
The angle passes through the point
(–1, 0) on the unit circle.
tan 180° =
Holt McDougal Algebra 2
Use tan θ =
.
10-3 The Unit Circle
Check It Out! Example 1c
Use the unit circle to find the exact value of
each trigonometric function.
The angle passes through the point
on the unit circle.
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.
Trigonometric Functions and Reference Angles
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 to find the exact value
of the sine, cosine, and tangent of 330°.
Step 1 Find the measure
of the reference angle.
The reference angle
measures 30°
Holt McDougal Algebra 2
10-3 The Unit Circle
Example 3 Continued
Step 2 Find the sine, cosine, and
tangent of the reference angle.
Use sin θ = y.
Use cos θ = x.
Holt McDougal Algebra 2
10-3 The Unit Circle
Example 3 Continued
Step 3 Adjust the signs, if needed.
In Quadrant IV, sin θ is
negative.
In Quadrant IV, cos θ is
positive.
In Quadrant IV, tan θ is
negative.
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 3a
Use a reference angle to find the exact value of
the sine, cosine, and tangent of 270°.
Step 1 Find the measure
of the reference angle.
The reference angle
measures 90°
Holt McDougal Algebra 2
270°
10-3 The Unit Circle
Check It Out! Example 3a Continued
Step 2 Find the sine, cosine, and
tangent of the reference angle.
sin 90° = 1
Use sin θ = y.
90°
cos 90° = 0
tan 90° = undef.
Holt McDougal Algebra 2
Use cos θ = x.
10-3 The Unit Circle
Check It Out! Example 3a Continued
Step 3 Adjust the signs, if needed.
sin 270° = –1
cos 270° = 0
tan 270° = undef.
Holt McDougal Algebra 2
In Quadrant IV, sin θ is
negative.
10-3 The Unit Circle
Check It Out! Example 3b
Use a reference angle to find the exact value of
the sine, cosine, and tangent of each angle.
Step 1 Find the measure
of the reference angle.
The reference angle
measures .
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 3b Continued
Step 2 Find the sine, cosine, and
tangent of the reference angle.
Use sin θ = y.
Use cos θ = x.
Holt McDougal Algebra 2
30°
10-3 The Unit Circle
Check It Out! Example 3b Continued
Step 3 Adjust the signs, if needed.
In Quadrant IV, sin θ is
negative.
In Quadrant IV, cos θ is
positive.
In Quadrant IV, tan θ is
negative.
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 3c
Use a reference angle to find the exact value of
the sine, cosine, and tangent of each angle.
–30°
Step 1 Find the measure
of the reference angle.
The reference angle
measures 30°.
Holt McDougal Algebra 2
–30°
10-3 The Unit Circle
Check It Out! Example 3c Continued
Step 2 Find the sine, cosine, and
tangent of the reference angle.
Use sin θ = y.
Use cos θ = x.
Holt McDougal Algebra 2
30°
10-3 The Unit Circle
Check It Out! Example 3c Continued
Step 3 Adjust the signs, if needed.
In Quadrant IV, sin θ is
negative.
In Quadrant IV, cos θ is
positive.
In Quadrant IV, tan θ is
negative.
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?
Step 1 Find the radius of the tire.
The radius is
diameter.
of the
Step 2 Find the angle θ through which the tire
rotates in 1 second.
Write a
proportion.
Holt McDougal Algebra 2
10-3 The Unit Circle
Example 4 Continued
The tire rotates θ radians in
1 s and 653(2) radians in
60 s.
Cross multiply.
Divide both sides by 60.
Simplify.
Holt McDougal Algebra 2
10-3 The Unit Circle
Example 4 Continued
Step 3 Find the length of the arc intercepted by
radians.
Use the arc length formula.
Substitute 0.325 for r and
for θ
Simplify by using a calculator.
The car travels about 22 meters in second.
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.
The radius is the actual
r =14
length of the hour hand.
Step 2 Find the angle θ through which the hour
hand rotates in 1 minute.
Write a
proportion.
Holt McDougal Algebra 2
10-3 The Unit Circle
Check It Out! Example 4 Continued
The hand rotates θ radians
in 1 m and 2 radians in
60 m.
Cross multiply.
Divide both sides by 60.
Simplify.
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.
Use the arc length formula.
s ≈ 1.5 feet
Substitute 14 for r and
for θ.
Simplify by using a calculator.
The minute hand travels about 1.5 feet in one minute.
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.
144°
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?
6.3 m
Holt McDougal Algebra 2