Transcript ch. 3

Copyright © 2005 Pearson Education, Inc.
Chapter 3
Radian Measure and
Circular Functions
Copyright © 2005 Pearson Education, Inc.
3.1
Radian Measure
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Radian Measure

An angle with its
vertex at the center
of a circle that
intercepts an arc on
the circle equal in
length to the radius
of the circle has a
measure of 1
radian.
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Slide 3-4
Converting Between Degrees and Radians


1. Multiply a degree measure by
simplify to convert to radians.

2. Multiply a radian measure by 180
and
simplify

to convert to degrees.
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180
radian and
Slide 3-5
Example: Degrees to Radians

Convert each degree measure to radians.
a) 60

b) 221.7

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Slide 3-6
Example: Radians to Degrees

Convert each radian measure to degrees.

a) 11
4

b) 3.25
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Slide 3-7
Equivalent Angles in Degrees and Radians
Degrees
Radians
Degrees
Exact
Approximate
0
0
0
30

6
45

4

3
60
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Radians
Exact
Approximate
90

2
1.57
.52
180

3.14
.79
270
3
2
4.71
1.05
360
2
6.28
Slide 3-8
Equivalent Angles in Degrees and Radians
continued
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Slide 3-9
Example: Finding Function Values of
Angles in Radian Measure



Find each function value.
4
a) tan
3

b) sin 4
3
Convert radians to
degrees.
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Slide 3-10
3.2
Applications of Radian Measure
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Arc Length

The length s of the arc
intercepted on a circle of
radius r by a central angle
of measure  radians is
given by the product of
the radius and the radian
measure of the angle, or
s = r,  in radians.
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Slide 3-12
Example: Finding Arc Length



A circle has radius 18.2
cm. Find the length of the
arc intercepted by a
central angle having each
of the following measures.
a) 3
8
b) 144
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Slide 3-13
Example: Finding a Length

A rope is being wound
around a drum with
radius .8725 ft. How
much rope will be wound
around the drum it the
drum is rotated through
an angle of 39.72?
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Slide 3-14
Example: Finding an Angle Measure

Two gears are adjusted
so that the smaller gear
drives the larger one, as
shown. If the smaller gear
rotates through 225,
through how many
degrees will the larger
gear rotate?
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Slide 3-15
Area of a Sector

A sector of a circle is a portion of the interior of
a circle intercepted by a central angle. “A piece
of pie.”

The area of a sector of a circle of radius r and
central angle  is given by
1 2
A  r ,
2
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 in radians.
Slide 3-16
Example: Area

Find the area of a sector with radius 12.7 cm
and angle  = 74.
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Slide 3-17
3.3
The Unit Circle and
Circular Functions
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Unit Circle



In the figure below, we start at the point (1, 0) and
measure an arc s along the circle. The end point of this
arc is (x, y). The circle is a unit circle - it has its center at
the origin and a radius of 1 unit.
For θ measured in radians, we know that s = r θ. Here,
r = 1, so s, is numerically equal to θ, measured in radians.
Thus, the trigonometric functions of angle θ in radians
found by choosing a point (x, y)
on the
unit circle can be rewritten
as functions
of the arc length s,
a real number.
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Slide 3-19
Circular Functions

sin s  y
cos s  x
y
tan s  ( x  0)
x
1
csc s  ( y  0)
y
1
sec s  ( x  0)
x
x
cot s  ( y  0)
y
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Slide 3-20
Unit Circle--more
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Slide 3-21
Domains of the Circular Functions


Assume that n is any integer and s is a real
number.
Sine and Cosine Functions: (, )


 

Tangent and Secant Functions: s | s   2n  1  
2 



Cotangent and Cosecant Functions: s | s  n 
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Slide 3-22
Evaluating a Circular Function

Circular function values of real numbers are
obtained in the same manner as trigonometric
function values of angles measured in radians.
This applies both to methods of finding exact
values (such as reference angle analysis) and to
calculator approximations. Calculators must be
in radian mode when finding circular function
values.
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Slide 3-23
Example: Finding Exact Circular
Function Values


7
7
7
Find the exact values of sin , cos , and tan .
4
4
4
Evaluating a circular function at the real number 7
7
4
is equivalent to evaluating it at 4 radians. An
7
angle of 4 intersects the unit circle at the point
 2  2 .
,

 2

2 
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Slide 3-24
20, 38
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Slide 3-25
Example: Approximating



Find a calculator approximation to four decimal
places for each circular function. (Make sure the
calculator is in radian mode.)
a) cos 2.01
b) cos .6207
For the cotangent, secant, and cosecant
functions values, we must use the appropriate
reciprocal functions.
c) cot 1.2071
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Slide 3-26
54, 58
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Slide 3-27
3.4
Linear and Angular Speed
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Angular and Linear Speed

Angular Speed: the amount of rotation per unit of
time, where  is the angle of rotation and t is the

time.

t

Linear Speed: distance traveled per unit of time
distance
s
speed =
or v  ,
time
t
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Slide 3-29
Formulas for Angular and Linear Speed
Angular Speed


t
(  in radians per unit
time,  in radians)
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Linear Speed
s
v
t
r
v
t
v  r
Slide 3-30
Example: Using the Formulas

Suppose that point P is on a circle with radius 20

cm, and ray OP is rotating with angular speed
18
radian per second.
a) Find the angle generated by P in 6 sec.
b) Find the distance traveled by P along the
circle in 6 sec.
c) Find the linear speed of P.
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Slide 3-31
Example: A belt runs a pulley of radius
6 cm at 80 revolutions per min.



a) Find the angular speed
of the pulley in radians
per second.
80(2) = 160  radians
per minute.
60 sec = 1 min
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

b) Find the linear speed
of the belt in centimeters
per second.
The linear speed of the
belt will be the same as
that of a point on the
circumference of the
pulley.
Slide 3-32
28, 40, 42
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Slide 3-33