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Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
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
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
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
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