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What you will learn

What radian measure is How to change from radian measure to

degree measure and vice versa How to find the length of an arc given

the measure of the central angle How to find the area of a sector

Objective: 6-1 Angles and Radian Measure

2Radian Measure

In most areas involving application of trigonometry, we use degree measures. In more advanced math work (and many sciences) radian measures are used. It allows real numbers to be used rather than degree measures.

Objective: 6-1 Angles and Radian Measure

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Radians?

1What is the circumference of a circle with a radius of 1?

Objective: 6-1 Angles and Radian Measure

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More Measurements

Objective: 6-1 Angles and Radian Measure

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Converting to Radians from Degrees

We have seen that:

2360 D

Therefore, to change from degrees to radians, multiply the degree measure by:

180

Objective: 6-1 Angles and Radian Measure

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Example Convert each degree measure to

radians:A. 45o

B. 115o

Objective: 6-1 Angles and Radian Measure

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Converting Radians to Degrees To convert radians to degrees:

In other words, multiply the radian measure by:

o180

3602 R

Objective: 6-1 Angles and Radian Measure

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Example Convert each radian measure to degrees.

A.

B.

4

9

8

7

Objective: 6-1 Angles and Radian Measure

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Finding Function Values for Angles in Radians

Our “special” angles:

Objective: 6-1 Angles and Radian Measure

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Evaluating Angles Measured in Radians Evaluate

You Try: Evaluate

3

4cos

6

25tan

Objective: 6-1 Angles and Radian Measure

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Circular Arcs and Central Angles Radian measure can be used to find the

length of a circular arc. A circular arc is a part of a circle. The arc is often defined by the central angle that intercepts it.

Objective: 6-1 Angles and Radian Measure

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Finding Arc Length The length of any circular arc s is equal to the

product of the measure of the radius of the circle r and the radian measure of the central angle that it subtends.

rs

Objective: 6-1 Angles and Radian Measure

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An Example Given a central angle of 128o, find the length of

the intercepted arc in a circle of radius 5 centimeters. Round to the nearest tenth.

Step 1: Convert degrees to radians.

Use to find the arc length. rs

Objective: 6-1 Angles and Radian Measure

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You Try Given a central angle of 125o, find the

length of its intercepted arc in a circle of radius 7 centimeters. Round to the nearest tenth.

Objective: 6-1 Angles and Radian Measure

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Application of Arc Length Winnipeg, Manitoba, Canada, and Dallas, Texas,

lie along the 97o W longitude line. The latitude of Winnipeg is 50o N, and the latitude of Dallas is 33o N. The radius of Earth is about 3960 miles. Find the approximate distance between the two cities.

Objective: 6-1 Angles and Radian Measure

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You Try The Swiss have long been highly regarded as

makers of fine watches. The central angle formed by the hands of the watch on “12” and “5” is 150o. The radius of the minute hand is ¾ centimeter. Find the distance traversed by the end of the minute hand from “12” to “5” to the nearest hundredth of a centimeter.

Objective: 6-1 Angles and Radian Measure

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Area of a Circular Sector If is the measure of the central angle

expressed in radians and r is the measure of the radius of the circle, then the area of the sector, A, is as follows:

22

1rA

Objective: 6-1 Angles and Radian Measure

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An Example Find the area of a sector if the central angle

measures radians and the radius of the circle is 16 centimeters. Round to the nearest tenth.

6

5

Objective: 6-1 Angles and Radian Measure

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You Try Find the area of a sector if the central angle

measures radians and the radius of the circle is 11 centimeters. Round to the nearest tenth.

7

3

Objective: 6-1 Angles and Radian Measure

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Homework Homework 1: page 348, 16-36 even, 40,

42.

Homework 2: page 349, 44-52 even, 55, 57 and challenge problem 54.