1 of 21 Pre Calculus Chapter 4.1 Warm - up. Chapter 4 Sec 1 Angles and Degree Measure.

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Pre Calculus Chapter 4.1

Warm - up

112112 .1 xx

223 .2 2 xxx

234

24

2

32 .3

12 .2

1214 .1

xxx

xx

x

completelyFactor

31 .3 2 xxx

Chapter 4 Sec 1

Angles and Degree Measure

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Essential Question

How do you describe angles and angular movement?

Key Vocabulary:Initial side

Terminal side

Linear speed

Angular speed

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• An angle in standard position has its vertex at the origin and initial side on the positive x–axis.

initial side

terminal side

Standard Position

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• Angles that have a counterclockwise rotation have a positive measure.

positive

0º or 2π

90º or π/2

180º or π

270º or 3π/2

Positively Counterclockwise

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• Angles that have a clockwise rotation have a negative measure.

Negative

Clockwise means negative

0º or –2π

– 270º or –3π/2

–180º or –π

–90º or – π/2

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• Coterminal angles are angles that have the same initial and terminal side, but differ by the number of rotations.

• Since one rotation equals 2π, the measures of coterminal angles differ by multiples of 2π.

Coterminal Angles

3

3

72

3

3

5

32

3

5

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Radian Measure

• The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.

• One way to measure angles is in radians.

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Radian…still

Since the circumference of a circle is 2πr units.

radians 2 2 rs

radians. 24

2srevolution

4

1

radians 2

2srevolution

2

1

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Pre Calculus Chapter 4.1 Example 1

a. For the positive angle subtract

2π to obtain a coterminal angle.

b. For the positive angle subtract

2π to obtain a coterminal angle.

c. For the negative angle add 2π

to obtain a coterminal angle.

6

13

4

3

2

6

13

6

2

4

3

4

5

3

2

2

3

2

3

4

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Pre Calculus Chapter 4.1 Complementary and Supplementary Angles

Two positive angles α and β are complementary if their sum is π/2 or 90°.

Two positive angles are supplementary if their sum is π or 180°.

Find the complement and supplement of the following

6 a.

6

5 b.complement supplement complement supplement

62

66

3

36

2

6

66

6

6

5

6

5

6

5

6

6

6

Because 5π/6 is greater than π/2 it has no complement. (Remember complements are positive angles.)

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Guided Practice

Determine two coterminal angles in radian measure (one positive and one negative) for each angle.

6 a.

3

2 b.

Find (if possible) the complement and supplement of the angle.

6

11,

6

13 3

4,

3

8

3 c.

4

3 d.

complement

supplement3

26

4

None

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• A second way to measure angle is in the terms of degree, denoted by °.

• One degree = 1/360 • To measure angles it is convenient to mark degrees on a

circle. So a full revolution is 360°, a half is 180° a quarter is 90°…

• We see 360° is one revolutionso 360°= 2π rad and 180° = π rad

Thus

Degree

180

rad 1 and rad 180

1

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Degree/Radian Conversion

degree180

Radians radians180

Degree

30°

45° 60°

90°

180° 360°

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Example 2

a. 135o

b. 540°

c. – 270°

d.

e.

f.

deg 180

rad deg 135

Convert from degrees to radians.

Convert from radians to degrees.

radians 4

3π

deg 180

rad deg 540

radians 3π

deg 180

rad deg 270

radians

2

3π

radians 2

π

radians 2

radians 2

9π

rad

deg 180rad

2

90

rad

deg 180rad 2

59.114

360

rad

deg 180rad

2

9

810

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Arc Length

Arc LengthFor a circle of radius r, a central angle θ intercepts an arc of length s is given bys = r θ Length of circular arc where θ is measured in radians. Note that if r = 1, then s = θ, and the radian measurement of θ equal the arc length

A circle has a radius of 4 inches. Find the length of the arc intercepted by central angle of 240°.

deg 180

rad deg 240240

radians 3

4

3

44

rs inches 16.76 3

16

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Pre Calculus Chapter 4.1 Linear and Angular Speed

Linear and Angular Speed

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 of the particle is:

Linear speed =

Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, then the angular speed of the particle is

Angular speed =

Linear speed measures how fast the particle moves and angular speed measures how fast the angle changes.

t

s

time

length arc

t

time

angle central

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Example 4

The second hand of a clock is 10.2 centimeters long. Find the linear speed of the tip of this second hand.

In one revolution, the arc length traveled is

The time required for one revolution of the second hand is t = 1 minute or 60 seconds

So the linear speed of the tip of the second hand is

rs 2 2.102 4.20

t

sspeedLinear cm/sec 07.1

60

4.20

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Example 5

A 15-inch diameter tire on a car makes 9.3 revolutions per second.a. Find the angular speed of the tire in radians per second.b. Find the linear speed of the car.

Because each revolution generates 2π radians, it follows that the tire turns (9.3)(2π)= 18.6π radians per second. So the angular speed is length traveled is

The linear speed of the tire is

secondper radians 6.18second 1

radians 6.18speedAngular

t

t

r

t

s speedLinear

in/sec 25.438

1

6.181521

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Essential Question

How do you describe angles and angular movement?

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Daily Assignment

• Chapter 4 Section 1 • Text Book

• Pg 265 – 267• #5 – 21 Mod4• #31 – 55 Mod4• #71 – 87 Mod4, #76

• Show all work for credit.

1 of 21 Pre Calculus Chapter 4.1 Warm - up. Chapter 4 Sec 1 Angles and Degree Measure.

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