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7.1 Angles and Their 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

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Angles

Trigonometry translated: _____________ of _____________

Angle Measure

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Standard Position

Vertex at origin

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

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Positive and negative angles

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

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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.

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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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Measuring Angles

1

360

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Acute and Obtuse Angles

Acute angles have measure between _____º and _____º.

Obtuse angles have measure between ____º and _____º.

Straight angles measure _______º.

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Angles are often classified according to the

QUADRANT in which their terminal sides lie.

Example:

50º is a ____ quadrant angle.

208º is a ____ quadrant angle. II I

-75º is a _____ quadrant angle. III IV

Classifying Angles

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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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1. Sketch in standard position. In which quadrant is located?

2. Sketch in standard position. In which quadrant is located?

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278.1

Sketching Angles (Degrees)

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Complementary and Supplementary Angles

Complementary Angles

Two positive angles are complementary if their sum is ______º

Angles that measure 22º and ____º are complements.

Supplementary Angles

Two positive angles are supplementary if their sum is _______º

Angles that measure 137º and ____º are supplements.

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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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Radian Measure

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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

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Conversions Between Degrees and Radians

Example

a) Convert from radians to degrees:

b) Convert from radians to degrees: 3.8

3

4

3

4

3.8

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Conversions Between Degrees and Radians

Try it!

c) Convert from degrees to radians (exact):

d) Convert from radians to degrees:13

6

13

6

675

675

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Conversions Between Degrees and Radians

Again!

e) Convert from degrees to radians (to 3 decimal places):

f) Convert from radians to degrees (to nearest tenth): 1 rad

5252

1

°

°

°

°

°

°

°

°

°

°

°

°

°

°

°

°

°

Degree and Radian Form of “Special” Angles

For a positive angle.

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1. Sketch in standard position. In which quadrant is located?

2. Sketch in standard position. In which quadrant is located?

2

3

Sketching Angles (Radians)

3

4

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1 minute (1’) = degree OR 1° = ______ minutes

1 second (1”) = _____ minute OR 1’ = _______ seconds

1 second (1”) = _____ degree OR 1° = ______ seconds

Example

Convert to decimal degrees: 52 15'42"

Degrees, minutes, and seconds

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Conversions between decimal degrees and degrees, minutes,seconds can be easily accomplished using your TI graphingcalculator.

The ANGLE menu on your calculator has built-in features for converting between decimal degrees and DMS.

Degrees, minutes, and seconds

Note that the seconds (“) symbol is not in the ANGLE menu.

Use for “ symbol.

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Using your TI graphing calculator,

Convert to decimal degrees to the nearest hundredth of a degree.

Convert 57.328° to an equivalent angle expressed to the nearest second.

14 32 '18"

Practice

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End of Section 7.1