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13-3The
13-3
TheUnit
UnitCircle
Circle
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
13-3 The Unit Circle
Objectives
Convert angle measures between
degrees and radians.
Find the values of trigonometric
functions on the unit circle.
Holt Algebra 2
13-3 The Unit Circle
Vocabulary
radian
unit circle
Holt Algebra 2
13-3 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 Algebra 2
13-3 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 Algebra 2
13-3 The Unit Circle
Holt Algebra 2
13-3 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 Algebra 2
13-3 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 Algebra 2
13-3 The Unit Circle
Check It Out! Example 1
Convert each measure from degrees to
radians or from radians to degrees.
a. 80°
b.
c. –36°
d. 4 radians
Holt Algebra 2
13-3 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 Algebra 2
13-3 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 Algebra 2
13-3 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 Algebra 2
Use cos θ = x.
13-3 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 Algebra 2
.
13-3 The Unit Circle
Check It Out! Example 1a
Use the unit circle to find the exact value of
each trigonometric function.
sin 315°
Holt Algebra 2
13-3 The Unit Circle
Check It Out! Example 1b
Use the unit circle to find the exact value of
each trigonometric function.
tan 180°
Holt Algebra 2
13-3 The Unit Circle
Check It Out! Example 1c
Use the unit circle to find the exact value of
each trigonometric function.
Holt Algebra 2
13-3 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 Algebra 2
13-3 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 Algebra 2
13-3 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 Algebra 2
13-3 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 Algebra 2
13-3 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 Algebra 2
13-3 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°.
Holt Algebra 2
13-3 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.
Holt Algebra 2
13-3 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°
Holt Algebra 2
13-3 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 Algebra 2
13-3 The Unit Circle
Holt Algebra 2
13-3 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 Algebra 2
13-3 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 Algebra 2
13-3 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 Algebra 2
13-3 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?
Holt Algebra 2