Radian Questions English

  1. A small coin with radius 1 cm is rolling around the inside edge of a much larger fixed coin with radius 10 cm. How many radians does the small coin rotate about its own center by the time it returns to its starting position?
  2. What is the exact radian measure of the angle at the center of a regular 17-sided polygon (heptadecagon) that is formed by connecting two non-adjacent vertices that are separated by exactly 5 sides?
  3. If you walk along a circular path of radius 8 meters and travel an arc length of exactly 10π meters, through how many full revolutions (expressed in radians) have you turned?
  4. The minute hand of a clock is 12 cm long. After how many minutes (starting from 12:00) will the tip of the minute hand have traveled exactly 25π cm along its circular path? Give the answer both in minutes and in radians swept by the hand.
  5. A goat is tied to a point on the circumference of a circular barn with radius r using a rope of length 2πr. What is the total area (in terms of r) that the goat can graze? (This is a classic — radians make the boundary description much cleaner.)
  6. Convert 1 degree per second into radians per hour. Then explain why this number looks surprisingly “clean” when expressed exactly.
  7. In radians, what is the period of the function f(x) = sin(√2 x) + cos(√3 x)? (Give a short reason why this question is trickier than it first appears.)
  8. A ladder 5 m long is sliding down a vertical wall. At the moment the bottom of the ladder is 3 m from the wall, how fast is the angle (in radians per second) between the ladder and the ground changing if the bottom is moving away at 2 cm/s?
  9. What is the smallest positive angle θ (in radians) such that sin(θ) = sin(θ + π/7) = sin(θ + 2π/7)? (This one connects to roots of unity and triple-angle behavior.)
  10. Imagine the unit circle. You start at (1,0) and walk along the circle at 1 radian per second. After t seconds, your coordinates are (cos t, sin t). At exactly what times t (in the first 100 seconds) will you be exactly as far from the point (0,1) as you are from the point (0,-1)? Give the general form.