Copyright © Cengage Learning. All rights reserved. 4 Trigonometry Copyright © Cengage Learning. All rights reserved.

Before Complete the following table:

Copyright © Cengage Learning. All rights reserved. 4.1 RADIAN AND DEGREE MEASURE Copyright © Cengage Learning. All rights reserved.

What You Should Learn Sec 4-1 Describe angles. Use radian measure. Use degree measure. Use angles to model and solve real-life problems.

Applications

Applications Begin completing section IV on page 75 of NTG. The radian measure formula,  = s / r, can be used to measure arc length along a circle. Begin completing section IV on page 75 of NTG.

Example 5 – Finding Arc Length A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240, as shown in Figure 4.15. Figure 4.15

Example 5 – Solution To use the formula s = r, first convert 240 to radian measure.

Example 5 – Solution cont’d Then, using a radius of r = 4 inches, you can find the arc length to be s = r Note that the units for r are determined by the units for r because  is given in radian measure, which has no units.

You Do A circle has a radius of 27 inches. Find the length of the arc intercepted by a central angle of 160°.

Applications Continue working on section IV page 75 NTG. The formula for the length of a circular arc can be used to analyze the motion of a particle moving at a constant speed along a circular path. Continue working on section IV page 75 NTG.

Example 6 The second hand of a clock is 10.2 cm long, as shown in the figure. Find the linear speed of the tip of this second hand as it passes around the clock face. In one complete revolution the second hand covers a distance of . The time required for this distance 1 minute or 60 seconds. Linear speed =

You Do: The second hand of a clock is 8 cm long, as shown in the figure. Find the linear speed of the tip of this second hand as it passes around the clock face. In one complete revolution the second hand covers a distance of . The time required for this distance 1 minute or 60 seconds. Linear speed =

Example 7 The blades of a wind turbine are 116 ft long. The propeller rotates at 15 revolutions per minute. Find the angular speed of the propeller in radians per minute. Find the linear speed of the tips of the blades. a) In one minute the blades make 15 revolutions, which is Angular speed = b) Linear speed =

You Do A circular saw blade rotates at 2400 revolutions per minute. Find the angular speed in radians per second. The blade has a radius of 4 in. Find the linear speed of a blade tip in inches per second.

Applications Complete the top of page 76 in NTG. A sector of a circle is the region bounded by two radii of the circle and their intercepted arc (see Figure 4.18). Figure 4.18 Complete the top of page 76 in NTG.

Applications

Example 8 A sprinkler on a golf course fairway sprays water over a distance of 70 ft and rotates through an angle of 120°. Find the area of the fairway watered by the sprinkler.

145° You Do a Mr. Green original. A sprinkler placed in front of the pitching rubber is set to cover the circular arc from foul line to foul line, and rotates through an angle of 145°. What is the area of the sector covered by the sprinkler? 145°

Exit Slip Suppose the sprinkler on the baseball field was set to rotate through the 145° in one minute, and you are at the infield/outfield grass line at first base. How fast would you have to travel (in feet per minute) to stay just ahead of the sprinkler? Linear speed =