AAT-A 4/25/14

Obj: SWBAT convert from degrees to radians and vice versa.Agenda•Bell Ringer: Inquiry: Angle measure•HW Requests: Comments on ACT•Turn in Break Packets•Tabled: Skills Practice WS Angle of Elevation and Depression•Chapter 8 Take Home Test•Homework: Read Section 13.2 #19-49 odds

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Larie, Midterm???

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6.1 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 Find coterminal angles

Angle-formed by rotating a ray about its endpoint (vertex)

Initial Side Starting position

Terminal Side Ending position

Standard Position Initial side on positive x-axis and the vertex is on the origin

An angle describes the amount and direction of rotation

120° –210°

Positive Angle- rotates counter-clockwise (CCW)

Negative Angle- rotates clockwise (CW)

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

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6.1 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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6.1 Radian and Degree Measure

Measuring Angles

1

360

7

In general, for in radians,

A second way to measure angles is in radians.

Radian Measure

s

r

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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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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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

Example

Convert from degrees to radians: 210º

210

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

Convert from radians to degrees.3.

4.

11

3

13

12

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6.1 Radian and Degree Measure

Coterminal Angles

Angles that have the same initial and terminal sides are

coterminal.

Angles and are coterminal.

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6.1 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

Coterminal Angles:Two angles with the same initial

and terminal sides

Find a positive coterminal angle to 20º 38036020

34036020Find a negative coterminal angle to 20º

Types of questions you will be asked:

Identify a) ALL angles coterminal with 45º, then b) find one positive coterminal angle and one negative coterminal angle.

a) 45º + 360k (where k is any given integer).

b) Some possible answers are 405º, 765º, - 315º, - 675º

To find a coterminal angle add or subtract multiples of 360º

if in degrees or 2π if in radians

Ex 5. Convert the degrees to radian measure.

a) 60

b) 30 c) d) -54

e) -118f)

g) 45

Class Work

a)

b)

c)

d)

6

2

11

18

9

15

Class Work Time permittingPg 712 #1, 2, 4-16

HW: Read Section 13.2 #19-49 odds

Ex 6. Convert the radians to degrees.

a)

b)

c)

d)

6

2

11

18

9

17

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

6.1 Radian and Degree Measure

Classifying Angles

Quadrant 1

Quadrant 3

Quadrant 4

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6.1 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.

Ex 3. Find one positive and one negative angle that is coterminal with the angle = 30° 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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6.1 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.

s

r

In general,

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6.1 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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6.1 Radian and Degree Measure

Radian Measure

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6.1 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

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.7

5

3

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0°

360 °

30 °

45 °

60 °

330 °

315 °

300 °

120 °

135 °

150 °

240 °

225 °

210 °

180 °

90 °

270 °

Degree and Radian Form of “Special” Angles

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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