Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 3 Radian Measure and the Unit Circle Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

3

Radian Measure and the Unit Circle



3.1 Radian Measure

3.2 Applications of Radian Measure

Radian Measure and Circular Functions3


Radian Measure3.1Radian Measure ▪ Converting Between Degrees and Radians ▪ Finding Function Values for Angles in Radians


Convert each degree measure to radians.

3.1 Example 1 Converting Degrees to Radians (page 95)


Convert each radian measure to degrees.

3.1 Example 2 Converting Radians to Degrees (page 95)


Find each function value.

3.1 Example 3 Finding Function Values of Angles in Radian Measure (page 97)


Applications of Radian Measure3.2Arc Length on a Circle ▪ Area of a Sector of a Circle


A circle has radius 25.60 cm. Find the length of the arc intercepted by a central angle having each of the following measures.

3.2 Example 1 Finding Arc Length Using s = rθ (page 101)

Convert θ to radians.


Erie, Pennsylvania is approximately due north of Columbia, South Carolina. The latitude of Erie is 42° N, while that of Columbia is 34° N. Find the north-south distance between the two cities.

3.2 Example 2 Using Latitudes to Find the Distance Between Two Cities (page 101)

The radius of the earth is about 6400 km.

The central angle between Erie and Columbia is 42° – 34° = 8°. Convert 8° to radians:


Use s = rθ to find the north-south distance between the two cities.

3.2 Example 2 Using Latitudes to Find the Distance Between Two Cities (cont.)

The north-south distance between Erie and Columbia is about 890 km.


A rope is being wound around a drum with radius 0.327 m. How much rope will be wound around the drum if the drum is rotated through an angle of 132.6°?

3.2 Example 3 Finding a Length Using s = rθ (page 102)

The length of rope wound around the drum is the arc length for a circle of radius 0.327 m and a central angle of 132.6°.


Use s = rθ to find the arc length, which is the length of the rope.

3.2 Example 3 Finding a Length Using s = rθ (cont.)

The length of the rope wound around the drum is about 0.757 m.

Convert 132.6° to radians:


Two gears are adjusted so that the smaller gear drives the larger one. If the radii of the gears are 3.6 in. and 5.4 in., and the smaller gear rotates through 150°, through how many degrees will the larger gear rotate?

3.2 Example 4 Finding an Angle Measure Using s = rθ

(page 102)

First find the radian measure of the angle, and then find the arc length on the smaller gear that determines the motion of the larger gear.


The arc length on the smaller gear is

3.2 Example 4 Finding an Angle Measure Using s = rθ

(cont.)

An arc with length 3π cm on the larger gear corresponds to an angle measure θ radians, where

The larger gear will rotate through 100°.


Find the area of a sector of a circle having radius 15.20 ft and central angle 108.0°.

3.2 Example 5 Finding the Area of a Sector (page 103)

The area of the sector is about 217.8 sq ft.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 3 Radian Measure and the Unit Circle Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1