Chapter 5Angles and Arcs in the Unit Circle

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5.1 Radian and Degree Measure

In this section, we will study the following topics:

Terminology used to describe angles Degree measure of an angle Radian measure of an angle Converting between radian and degree

measure Finding coterminal angles

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5.1 Radian and Degree Measure

Angles

Trigonometry: measurement of triangles

Angle Measure

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Section 4.1, Figure 4.1, Terminal and Initial Side of an Angle , pg. 248

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5.1 Radian and Degree Measure

Standard Position

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Section 4.1, Figure 4.2, Standard Position of an Angle, pg. 248

Vertex at origin

The initial side of an angle in standard position is always located on the positive x-axis.

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6.1 Radian and Degree Measure

Positive and negative angles

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Section 4.1, Figure 4.3, Positive and Negative Angles, pg. 248

When sketching angles, always use an arrow to show direction.

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5.1 Radian and Degree Measure

Measuring Angles

The measure of an angle is determined by the amount of

rotation from the initial side to the terminal side.

There are two common ways to measure angles, in degrees

and in radians.

We’ll start with degrees, denoted by the symbol º.

One degree (1º) is equivalent to a rotation of of one

revolution.

1

360

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5.1 Radian and Degree Measure

Measuring Angles

1

360

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Section 4.1, Figure 4.13, Common Degree Measures on the Unit Circle, pg. 251

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5.1 Radian and Degree Measure

Angles are often classified according to the

quadrant in which their terminal sides lie.

Ex1: Name the quadrant in which each angle

lies.

50º

208º II I

-75º III

IV

Classifying Angles

Quadrant 1

Quadrant 3

Quadrant 4

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5.1 Radian and Degree Measure

Classifying Angles

Standard position angles that have their terminal side

on one of the axes are called quadrantal angles.

For example, 0º, 90º, 180º, 270º, 360º, … are

quadrantal angles.

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5.1 Radian and Degree Measure

Coterminal Angles

Angles that have the same initial and terminal sides are

coterminal.

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Section 4.1, Figure 4.4, Coterminal Angles, pg. 248

Angles and are coterminal.

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5.1 Radian and Degree Measure

Example of Finding Coterminal Angles

You can find an angle that is coterminal to a given angle by

adding or subtracting multiples of 360º.

Ex 2:

Find one positive and one negative angle that are

coterminal to 112º.

For a positive coterminal angle, add 360º : 112º + 360º = 472º

For a negative coterminal angle, subtract 360º: 112º - 360º = -248º

Ex. 3: Find one positive and one negative angle that is coterminal with the angle = 30 o

in standard position.

Ex. 4: Find one positive and one negative angle that is coterminal with the angle =

272 in standard position.

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5.1 Radian and Degree Measure

Radian Measure

A second way to measure angles is in radians.

Definition of Radian:

One radian is the measure of a central angle that intercepts arc s equal in length to the radius r of the circle.

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Section 4.1, Figure 4.5, Illustration of Arc Length, pg. 249

s

r

In general,

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5.1 Radian and Degree Measure

Radian Measure

2 radians corresponds to 360

radians corresponds to 180

radians corresponds to 902

2 6.28

3.14

1.572

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Section 4.1, Figure 4.6, Illustration of Six Radian Lengths, pg. 249

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5.1 Radian and Degree Measure

Radian Measure

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Section 4.1, Figure 4.7, Common Radian Angles, pg. 249

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5.1 Radian and Degree Measure

Conversions Between Degrees and Radians

1. To convert degrees to radians, multiply degrees by

2. To convert radians to degrees, multiply radians by

180

180

a) 60

b) 30

c) -54

d) -118

e) 45

Ex 5. Convert the degrees to radian measure.

a)

b)

c)

d)

e)

Ex 6. Convert the radians to degrees.

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Ex 7. Find one positive and one negative angle that is coterminal with the angle = in standard position.

Ex 8: Find one positive and one negative angle that is coterminal with the angle = in standard position.

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3

20

0°

360 °

30 °

45 °

60 °

330 °

315 °

300 °

120 °

135 °

150 °

240 °

225 °

210 °

180 °

90 °

270 °

Degree and Radian Form of “Special” Angles

Convert from degrees to radians.1. 542. -300

Convert from radians to degrees.3.

4.

Class Work

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Find one postive angle and one negative angle in standard position that are coterminal with the given angle.

5. 135

6. 11

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