Find the measure of each angle in degrees and radians.

Problem 9

This is not correct. It would be correct if the longer arc mark was not there. However, because it is there, you have to travel 360-30 = 330 degrees in the negative direction. So the angle shown along that arc is -330 degrees which converts to -11pi/6 radians.

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Problem 10

The spiral arc almost wraps itself around 2 times over, effectively almost making two full revolutions.

1 revolution = 2pi radians

2 revolutions = 4pi radians (multiply both sides by 2)

Subtract off pi/3

4pi - pi/3 = 12pi/3 - pi/3 = 11pi/3

11pi/3 represents the radian measure of the spiral arc

Convert to degrees

11pi/3 radians = (11pi/3)*(180/pi)

11pi/3 radians = (11/3)*180

11pi/3 radians = 11*180/3

11pi/3 radians = 11*60

11pi/3 radians = 660 degrees

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Problem 11

Add pi/2 radians to the angle measure shown

pi/2 + 7pi/18 = 9pi/18 + 7pi/18

pi/2 + 7pi/18 = 16pi/18

pi/2 + 7pi/18 = 8pi/9

The full angle in radians is 8pi/9 radians

Now convert to degrees

8pi/9 radians = (8pi/9)*(180/pi)

8pi/9 radians = (8/9)*180

8pi/9 radians = 8*180/9

8pi/9 radians = 8*20

8pi/9 radians = 160 degrees

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Problem 12

Add pi/2 radians (aka 90 degrees) to pi/9 to get

pi/2 + pi/9 = 9pi/18 + 2pi/18

pi/2 + pi/9 = 11pi/18

This represents the red angle shown in the diagram below.

Subtract this found angle from 4pi, which represents 2 full revolutions

4pi - 11pi/18 = 72pi/18 - 11pi/18

4pi - 11pi/18 = 61pi/18

The spiral mark shown in the diagram is 61pi/18 radians

Convert to degrees

61pi/18 radians = (61pi/18)*(180/pi)

61pi/18 radians = (61/18)*180

61pi/18 radians = 61*180/18

61pi/18 radians = 61*10

61pi/18 radians = 610 degrees