Radians and Degrees

Show your work for each problem.

1.
How many degrees are in a full circle?
1. a)
2. b)
3. c)
4. d)
Answer:
A full circle is . Note that is also a full circle, but measured in radians, not degrees.
2.
What is the key relationship between degrees and radians?
1. a) radians
2. b) radians
3. c) radians
4. d) radians
Answer: radians
The fundamental relationship is radians. This is half a circle. From this, we get and .
3.
Convert to radians. Express your answer as a fraction of (e.g., pi/2).
Answer: pi/2
radians. This is a quarter of a circle.
4.
Convert radians to degrees.
Answer: 30
. The cancels out, leaving a simple division.
5.
Convert to radians.
1. a)
2. b)
3. c)
4. d)
Answer:
. Note: is , is , and is .
6.
Convert to radians and simplify.
Answer: 3pi/4
1. What formula do you use to convert degrees to radians? degrees * pi/180
2. Set up the multiplication: = ? 135pi/180
3. Find the GCD of 135 and 180 to simplify. 45
4. Divide both by 45: 3pi/4
7.
Convert radians to degrees.
Answer: 225
. This angle is in the third quadrant.
8.
Which of these is closest to 1 radian in degrees?
1. a)
2. b)
3. c)
4. d)
Answer:
. One radian is the angle where the arc length equals the radius.
9.
Convert radians to degrees and identify the quadrant.
Answer: 210, third
1. Set up the conversion: . The cancels. What is ? 210
2. Is between 0° and 90°, 90° and 180°, 180° and 270°, or 270° and 360°? 180 and 270
3. Which quadrant is this? third
10.
Convert to radians. Express as a fraction of .
Answer: 3pi/2
radians. This is three-quarters of a full circle.
11.
A wheel rotates at radians per second. How many complete rotations does it make per second?
1. a)4
2. b)2
3. c)
4. d)1
Answer: 2
One rotation = rad. So rotations per second.
12.
Convert to radians and find its reference angle.
Answer: 7pi/4, 45
1. Convert to radians: 315pi/180
2. Simplify by finding the GCD of 315 and 180. What is the GCD? 45
3. Simplify to lowest terms. 7pi/4
4. is in which quadrant? fourth
5. For fourth quadrant, reference angle = 45
13.
Convert radians to degrees.
Answer: 330
. This angle is before a full rotation.
14.
Convert to radians. Express your answer as a simplified fraction of .
Answer: -pi/4
1. Does the formula change for negative angles? no
2. Calculate -45pi/180
3. Simplify by dividing by 45 -pi/4
15.
Which angle is greater: radians or ?
1. a)
2. b)Cannot be determined
3. c) radians
4. d)They are equal
Answer: radians
. Since , radians is greater.
16.
A pendulum swings through an arc of radians. What is this angle in degrees?
Answer: 36
1. What formula converts radians to degrees? radians * 180/pi
2. Set up: . What happens to ? cancels
3. After canceling , what is left? 180/5
4. Calculate 36