3 Radian Measure and the Unit Circle Copyright

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

3 Radian Measure and Circular Functions 3. 1 Radian Measure 3. 2 Applications of Radian Measure Copyright © 2013, 2009, 2005 Pearson Education, Inc. 2

3. 1 Radian Measure ▪ Converting Between Degrees and Radians ▪ Finding Function Values for Angles in Radians Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3

3. 1 Example (page 95) 1 Converting Degrees to Radians Convert each degree measure to radians. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4

3. 1 Example (page 95) 2 Converting Radians to Degrees Convert each radian measure to degrees. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5

3. 1 Example 3 Finding Function Values of Angles in Radian Measure (page 97) Find each function value. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6

3. 2 Applications of Radian Measure Arc Length on a Circle ▪ Area of a Sector of a Circle Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7

3. 2 Example 1 Finding Arc Length Using s = rθ (page 101) A circle has radius 25. 60 cm. Find the length of the arc intercepted by a central angle having each of the following measures. Convert θ to radians. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8

3. 2 Example 2 Using Latitudes to Find the Distance Between Two Cities (page 101) 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. The radius of the earth is about 6400 km. The central angle between Erie and Columbia is 42° – 34° = 8°. Convert 8° to radians: Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9

3. 2 Example 2 Using Latitudes to Find the Distance Between Two Cities (cont. ) Use s = rθ to find the north-south distance between the two cities. The north-south distance between Erie and Columbia is about 890 km. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10

3. 2 Example 3 Finding a Length Using s = rθ (page 102) 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°? 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°. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11

3. 2 Example 3 Finding a Length Using s = rθ (cont. ) Convert 132. 6° to radians: Use s = rθ to find the arc length, which is the length of the rope. The length of the rope wound around the drum is about 0. 757 m. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12

3. 2 Example 4 Finding an Angle Measure Using s = rθ (page 102) 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? 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. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13

3. 2 Example 4 Finding an Angle Measure Using s = rθ (cont. ) The arc length on the smaller gear is 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°. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14

3. 2 Example 5 Finding the Area of a Sector (page 103) Find the area of a sector of a circle having radius 15. 20 ft and central angle 108. 0°. The area of the sector is about 217. 8 sq ft. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15