Name_____________________________________ Class____________________________ Date________________
Lesson 13-3
Radian Measure
Lesson Objectives
1 Using radian measure for angles
Finding the length of an arc of a circle
2
Local Standards: ____________________________________
Vocabulary and Key Concepts.
Converting Between Radians and Degrees
All rights reserved.
To convert degrees to radians, multiply by p radians
1808 .
1808 .
To convert radians to degrees, multiply by p radians
Length of an Intercepted Arc
r
For a circle of radius r and a central angle of measure θ
(in radians), the length s of the intercepted arc is s θ
.
s
r
An intercepted arc of a circle is
A radian is
r
r
258
Algebra 2 Lesson 13-3
Daily Notetaking Guide
© Pearson Education, Inc., publishing as Pearson Prentice Hall.
A central angle of a circle is
Name_____________________________________ Class____________________________ Date ________________
Example.
1 Using a Proportion
a. Find the radian measure of an angle of 45.
°
°
πr radians
radians
Write a proportion.
? π ?
r
All rights reserved.
Write the cross-products.
? r
Divide each side by
.
Simplify.
An angle of 45 measures about
radians.
b. Find the degree measure of 13p
6 radians.
π radians
d°
180
Write a proportion.
© Pearson Education, Inc., publishing as Pearson Prentice Hall.
? 180 π ? d
d
Write the cross-products.
π ? 180
6 ? π
An angle of 13p
6 radians measures
Divide each side by
.
Simplify.
degrees.
Quick Check.
1. Use a proportion for each conversion.
a. 85 to radians
Daily Notetaking Guide
b. 2.5 radians to degrees
Algebra 2 Lesson 13-3
259
Name_____________________________________ Class____________________________ Date ________________
Examples.
2 Finding Cosine and Sine of Radian Measures Find the exact
1
values of cos (p3 radians) and sin (p3 radians).
°
°
⫽
Convert radians to degrees.
x
1
O
Draw the angle. Complete a 30-60-90 triangle.
The hypotenuse has length
leg is
1
.
the length of the hypotenuse, and the longer
All rights reserved.
The shorter leg is
1
60°
times the length of the shorter leg.
Thus, cos ( p
3 radians) and sin (p3 radians) .
3 Finding the Length of an Arc Use the circle at the right.
Find length s to the nearest tenth.
Use the formula.
s rθ
?
Substitute
for r and
Simplify.
Use a calculator.
The arc has length
s
7π
6
for θ.
b
π
3
6 in.
in.
Quick Check.
p
2. a. Use a calculator to find cos (p
3 radians) and sin ( 3 radians). How do these
values compare to the values found in Example 2?
b. Explain how to use mental math to convert p
3 radians to degrees.
3. Find length b in Example 3. Round your answer to the nearest tenth of an inch.
260
Algebra 2 Lesson 13-3
Daily Notetaking Guide
© Pearson Education, Inc., publishing as Pearson Prentice Hall.
π radians
?
3
y