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4Trigonometry

2

Before

Complete the following table:

degrees 30 45 60 90 120 135 150 180radiansdegrees 210 225 240 270 300 315 330 360radians

4.1 RADIAN AND DEGREE MEASURE

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4

• Describe angles.

• Use radian measure.

• Use degree measure.

• Use angles to model and solve real-life problems.

What You Should Learn Sec 4-1

5

Applications

6

Applications

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.

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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

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Example 5 – Solution

To use the formula s = r, first convert 240 to radian measure.

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Example 5 – Solution

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.

cont’d

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You Do

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

160 81 180 9

827 24 inches

975.40 inches

s r

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Applications

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.

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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 . 2 10.2 20.4 cm

The time required for this distance 1 minute or 60 seconds.

Linear speed =st

20.41.068 cm/sec

60

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8

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 . 2 8 16 cm

The time required for this distance 1 minute or 60 seconds.

Linear speed =st

160.838 cm/sec

60

You Do:

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Example 7

The blades of a wind turbine are 116 ft long. The propeller rotates at 15 revolutions per minute.a) Find the angular speed of the propeller in radians per

minute.b) Find the linear speed of the tips of the blades.

In one minute the blades make 15 revolutions, which is 2 15 30 rad/min

Angular speed =

30 rad30 rad / min

1 min

t

Linear speed =s rtt

116 30 ft3480 ft/min

1 min10,932.74 ft/min

a)

b)

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You Do

A circular saw blade rotates at 2400 revolutions per minute.a) Find the angular speed in radians per second.

b) The blade has a radius of 4 in. Find the linear speed of a blade tip in inches per second.

2400 rpm40 rps

60 sec

angular speed

2 401 sec

80 rad/sec

t

linear speed

4 801 sec

320 in/sec1005.31 in/sec

s rtt

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Applications

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.

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Applications

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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.

120 21 180 3

2

2

2

12

1 270

2 34900

35131.27 ft

A r

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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?

You Do a Mr. Green original.

2

2

2

145 291 180 36

12

1 2995

2 3611,419.9 ft

A r

145°

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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 =

2995

361

240.42 ft/min

s rtt