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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
