lat03 03

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Chapter 3

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3.1

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Radian Measure



An angle with its

vertex at the center

of a circle that

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Converting Between Degrees and Radians

 1. Multiply a degree measure by radian

and

simplify to convert to radians.

 2. Multiply a radian measure by and

simplify

to convert to degrees.

180





180



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Example: Degrees to Radians

 Convert each degree measure to radians.  a) 60

 b) 221.7

60 60 radia

80 n

1 3



 

 _ _ 

  



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Example: Radians to Degrees

 Convert each radian measure to degrees.

 a)

 b) 3.25

11 4

 11 11 180

495

4 4 

     _ _      180

3.25 3.25 186.2

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Example: Finding Function Values of

Angles in Radian Measure

 Find each function value.  a)

 Convert radians to

degrees.  b) 4 tan 3  180 4 4 tan tan 3 3 tan 240  _  _         4 sin 3  4

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3.2

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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

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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)

 b) 144 3

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Example: Finding Arc Length continued

 a) r = 18.2 cm and  =



 b) convert 144 to radians 3 8  cm 54.6 cm 21 1 .4cm 8 2 3 8. 8 s r s s        _ _     144 180 radi 144 an 4 5 s      _ _     cm 72.8

18.2 4

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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?  Convert 39.72 to radian

measure.

180

.8725 39.72 .6049 ft.

s r

s 





 _ _

 _ _ __ 

 

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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

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Solution

 Find the radian measure of the angle and then

find the arc length on the smaller gear that determines the motion of the larger gear.

5 225 225

180 4 5 12.5

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Solution continued

 An arc with this length on the larger gear

corresponds to an angle measure , in radians where

 Convert back to degrees.

4.8 125 19 8 2 25

s r

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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

2

1

, in radians. 2

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Example: Area

 Find the area of a sector with radius 12.7 cm

and angle  = 74.

 Convert 74 to radians.

 Use the formula to find the area of the sector of

a circle.

74 radian

74 s

180 1.2 29 

 

 _ _ 

 



2 2_1.29 2

1 1

(12.7) 2 104.193 cm

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3.3

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Unit Circle

 A unit circle has its center at the origin and a

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Circular Functions

 _sin _cos _tan ₍ ₀₎

1 1

csc ( 0) sec ( 0) cot ( 0)

y

s y s x s x

x

s y s x s y

y x y

   

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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:

 Cotangent and Cosecant Functions:

| 2 1

2

s s n 

 _ _ _ _

 _ _

 

 

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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 methods of finding exact

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Example: Finding Exact Circular

Function Values

 Find the exact values of

 Evaluating a circular function at the real number is

equivalent to evaluating it at radians. An angle of intersects the unit circle at the point .

 Since sin s = y, cos s = x, and



7 7 7

sin , cos , and tan .

4 4 4

   7 4  2 2 2 2

7 2 7 2 7

sin cos tan 1

4 2 4 2 4

          7 4  7 4  2 2 , 2 2  _     

tan s y x

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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  .4252 b) cos .6207

 .8135

 For the cotangent, secant, and cosecant functions

values, we must use the appropriate reciprocal

functions. _cot1.2071 1 _.3806

tan1.2071

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3.4

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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.

 Linear Speed: distance traveled per unit of time t

  

distance

speed = or

time  ,

s v

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Formulas for Angular and Linear Speed

( in radians per unit time,  in radians)

Linear Speed Angular Speed

t 

  v s_t

r v

t v r







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Example: Using the Formulas

 Suppose that point P is on a circle with radius 20

cm, and ray OP is rotating with angular speed

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.

18

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Solution: Find the angle.

 The speed of ray OP is radian per

second. 18

  

18

6

radians. 1

6

3

8

 

 



 

 

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Solution: Find the angle continued

 The distance traveled by P along the circle is

20 20 cm

3

3 .

 

 

   _{ } 

 

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Solution: Find the angle continued

 linear speed

20

20 1 10

6 cm per sec

3

9

2

6

0

3 



 



 

 

s

v

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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

 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.

160

radians per sec 6 8 3 0     