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Radian and Degree Measure
MATH 160, Precalculus
J. Robert Buchanan
Department of Mathematics
Fall 2011
J. Robert Buchanan
Radian and Degree Measure
Objectives
In this lesson we will learn to:
describe angles,
use radian measure,
use degree measure,
use angles to model and solve real-world problems.
J. Robert Buchanan
Radian and Degree Measure
Background
We now begin a study of trigonometry (Greek for
“measurement of triangles”).
Terminal Side
Vertex
Initial Side
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Radian and Degree Measure
Standard Position
If the vertex is the origin and the initial side lies along the
positive x-axis, the angle is said to be in standard position.
y
Terminal Side
x
Initial Side
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Radian and Degree Measure
Orientation and Notation
Angles will be named by uppercase letters (A, B, C, . . . ) or
Greek letters (α, β, γ, . . . ).
An angle is positive if it is generated by rotating the
terminal side counterclockwise (abbreviated ccw) from
the initial side.
An angle is negative if it is generated by rotating the
terminal side clockwise (abbreviated cw) from the initial
side.
Angles that have the same initial and terminal sides are
called coterminal angles.
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Radian and Degree Measure
Radian Measure
Definition
One radian is the measure of a central angle θ that intercepts
an arc s equal in length to the radius r of the circle. If θ is the
radian measure of the angle then
θ=
s
.
r
Recall: the circumference of a circle is C = 2πr , thus the
radian measure of one complete revolution is
θ=
2πr
= 2π ≈ 6.28.
r
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Radian and Degree Measure
Illustration
y
r
r=s
Θ
x
r
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Radian and Degree Measure
Common Angles (in radians)
А6
А4
А2
Π
А3
3А2
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Radian and Degree Measure
Angles and Quadrants
Θ=А2
Quadrant II
Quadrant I
А2<Θ<Π
0<Θ<А2
Θ=Π
Θ=0
Π<Θ<3А2
3А2<Θ<2Π
Quadrant III
Quadrant IV
Θ=3А2
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Radian and Degree Measure
Coterminal Angles
Recall: two angles are coterminal if they have the same initial
and terminal sides.
Since there are 2π radians in a complete revolution, coterminal
angles will have radian measures which differ by an integer
multiple of 2π.
Angles α and β are coterminal if they are in standard position
and
α = β + 2n π
for some integer n.
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Radian and Degree Measure
Examples
Find three coterminal angles to each of the following.
π
θ=−
6
2π
θ=
3
5π
θ=−
4
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Radian and Degree Measure
Complementary and Supplementary Angles
Definition
Two positive angles α and β are complementary if their sum is
π/2. Two positive angles α and β are supplementary if their
sum is π.
J. Robert Buchanan
Radian and Degree Measure
Complementary and Supplementary Angles
Definition
Two positive angles α and β are complementary if their sum is
π/2. Two positive angles α and β are supplementary if their
sum is π.
Example
If possible, find the complement and the supplement to each of
the following angles.
3π
θ=
7
θ=
2π
3
J. Robert Buchanan
Radian and Degree Measure
Complementary and Supplementary Angles
Definition
Two positive angles α and β are complementary if their sum is
π/2. Two positive angles α and β are supplementary if their
sum is π.
Example
If possible, find the complement and the supplement to each of
the following angles.
3π
θ=
7
π
4π
Complement:
, Supplement:
14
7
2π
θ=
3
J. Robert Buchanan
Radian and Degree Measure
Complementary and Supplementary Angles
Definition
Two positive angles α and β are complementary if their sum is
π/2. Two positive angles α and β are supplementary if their
sum is π.
Example
If possible, find the complement and the supplement to each of
the following angles.
3π
θ=
7
π
4π
Complement:
, Supplement:
14
7
2π
θ=
3
π
Complement: none, Supplement:
3
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Radian and Degree Measure
Degree Measure
Degrees are another way to measure angles. A degree is
equivalent to a rotation of 1/360 of a complete revolution.
Conversions
1
2
π rad
.
180◦
180◦
To convert radians to degrees, multiply radians by
.
π rad
To convert degrees to radians, multiply degrees by
J. Robert Buchanan
Radian and Degree Measure
Illustration
90°
°
60°
120
135°
45°
150°
30°
180°
0°
210°
330°
225°
315°
240°
300°
270°
J. Robert Buchanan
Radian and Degree Measure
Examples
Complete the angle measures in the following table.
Angle
α
β
γ
θ
Radian Measure
Degree Measure
15◦
π/3
135◦
5π/3
J. Robert Buchanan
Radian and Degree Measure
Examples
Complete the angle measures in the following table.
Angle
α
β
γ
θ
Radian Measure
π/12
π/3
3π/4
5π/3
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Degree Measure
15◦
60◦
135◦
300◦
Radian and Degree Measure
Application: Arc Length
Arc Length
For a circle of radius r , a central angle θ intercepts an arc of
length s given by
s = rθ
where θ is measured in radians. Note that if r = 1, then s = θ,
and the radian measure of θ equals the arc length.
J. Robert Buchanan
Radian and Degree Measure
Example
A circle has a radius of 5 feet. Find the length of the arc
intercepted by a central angle of
5π/4
120◦
J. Robert Buchanan
Radian and Degree Measure
Example
A circle has a radius of 5 feet. Find the length of the arc
intercepted by a central angle of
5π/4
s = (5)
5π
25π
=
≈ 19.635
4
4
feet
120◦
J. Robert Buchanan
Radian and Degree Measure
Example
A circle has a radius of 5 feet. Find the length of the arc
intercepted by a central angle of
5π/4
s = (5)
5π
25π
=
≈ 19.635
4
4
feet
120◦
s = (5)(120◦ )
π
10π
=
≈ 10.472
180◦
3
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Radian and Degree Measure
feet
Application: Linear and Angular Speeds
Linear and Angular Speeds
Consider a particle moving at a constant speed along a circular
arc of radius r . If s is the length of the arc traveled in time t,
then the linear speed v of the particle is
Linear speed v =
s
arc length
= .
time
t
Moreover, if θ is the angle (in radian measure) corresponding to
the arc length s, then the angular speed ω (lowercase Greek
letter omega) of the particle is
Angular speed ω =
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central angle
θ
= .
time
t
Radian and Degree Measure
Example
A carousel with a 50-foot diameter makes 4 revolutions per
minute.
1
Find the angular speed of the carousel in radians per
minute.
2
Find the linear speed (in feet per minute) of the platform
rim of the carousel.
J. Robert Buchanan
Radian and Degree Measure
Example
A carousel with a 50-foot diameter makes 4 revolutions per
minute.
1
Find the angular speed of the carousel in radians per
minute.
ω = 4 rev/min = (4 rev/min)(2π rad/rev) = 8π rad/min
2
Find the linear speed (in feet per minute) of the platform
rim of the carousel.
J. Robert Buchanan
Radian and Degree Measure
Example
A carousel with a 50-foot diameter makes 4 revolutions per
minute.
1
Find the angular speed of the carousel in radians per
minute.
ω = 4 rev/min = (4 rev/min)(2π rad/rev) = 8π rad/min
2
Find the linear speed (in feet per minute) of the platform
rim of the carousel.
v = r ω = (25)(8π) = 200π ≈ 828.32 feet/min
J. Robert Buchanan
Radian and Degree Measure
Application: Area of a Sector
A sector of a circle is the region bounded between the two radii
of an intercepted arc.
Θ
r
A=
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1 2
r θ
2
Radian and Degree Measure
Example
A car’s rear windshield wiper rotates 125◦ . The total length of
the wiper mechanism is 25 inches and the blade wipes the
windshield over a distance of 14 inches. Find the area covered
by the wiper.
J. Robert Buchanan
Radian and Degree Measure
Example
A car’s rear windshield wiper rotates 125◦ . The total length of
the wiper mechanism is 25 inches and the blade wipes the
windshield over a distance of 14 inches. Find the area covered
by the wiper.
First convert the angle to radian measure.
125◦ =
125π
25π
=
180
36
J. Robert Buchanan
Radian and Degree Measure
Example
A car’s rear windshield wiper rotates 125◦ . The total length of
the wiper mechanism is 25 inches and the blade wipes the
windshield over a distance of 14 inches. Find the area covered
by the wiper.
First convert the angle to radian measure.
125π
25π
=
180
36
1
25π
1
25π
A =
(25)2
− (11)2
2
36
2
36
= 175π ≈ 549.779 in2
125◦ =
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Radian and Degree Measure
Homework
Read Section 4.1.
Exercises: 1, 5, 9, 13, . . . , 113, 117
J. Robert Buchanan
Radian and Degree Measure