Lesson 10

Beyond $2\pi$

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Problem 1

A rotation takes \(P\) to \(Q\). What could be the measure of the angle of rotation in radians? Select all that apply.

Circle on an x y axis. Point P is on the circle on the positive x axis. Point Q is on the circle on the positive y axis.
A:

\(\frac{3\pi}{2}\)

B:

\(\frac{\pi}{2}\)

C:

\(\frac{\pi}{4}\)

D:

\(\frac{5\pi}{2}\)

E:

\(\frac{5\pi}{4}\)

Solution

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Problem 2

  1. A \(\frac{2\pi}{3}\) radian rotation takes \(N\) to \(P\). Label \(P\).
  2. A \(\frac{7\pi}{6}\) radian rotation takes \(N\) to \(Q\). Label \(Q\).
  3. A \(\frac{25\pi}{6}\) radian rotation takes \(N\) to \(R\). Label \(R\).
Circle on an x y axis. Point N on the circle on the positive x axis.

Solution

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Problem 3

Here is a wheel with radius 1 foot.

A circle with center at the origin of an x y plane. 
  1. List three different counterclockwise angles the wheel can rotate so that point \(P\) ends up at position \(Q\).
  2. How many feet does the wheel roll for each of these angles?

Solution

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Problem 4

The point \(P\) on the unit circle is in the 0 radian position.

  1. Which counterclockwise rotations take \(P\) back to itself? Explain how you know.
  2. Which counterclockwise rotations take \(P\) to the opposite point on the unit circle? Explain how you know.

Solution

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Problem 5

Here is the unit circle with a point \(P\) at \((1,0)\). Find the coordinates of \(P\) after the circle rotates the given amount counterclockwise around its center.

A circle with center O at the origin of an x y plane. Point P lies on the outside of the circle, on the x axis, to the right of the origin.
  1. \(\frac{1}{3}\) of a full rotation
  2. \(\frac{1}{2}\) of a full rotation
  3. \(\frac{2}{3}\) of a full rotation

Solution

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(From Unit 6, Lesson 4.)

Problem 6

Here is a graph of \(y = \sin(\theta)\).

  1. Plot the points on the graph where \(\sin(\theta) = \text-\frac{1}{2}\).
  2. For which angles \(\theta\) does \(\sin(\theta) = \text-\frac{1}{2}\)?
Sine graph. 

Solution

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(From Unit 6, Lesson 9.)