# Lesson 10

Beyond $2\pi$

### Problem 1

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

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

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

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

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

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

### Solution

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### Problem 2

- A \(\frac{2\pi}{3}\) radian rotation takes \(N\) to \(P\). Label \(P\).
- A \(\frac{7\pi}{6}\) radian rotation takes \(N\) to \(Q\). Label \(Q\).
- A \(\frac{25\pi}{6}\) radian rotation takes \(N\) to \(R\). Label \(R\).

### Solution

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### Problem 3

Here is a wheel with radius 1 foot.

- List three different counterclockwise angles the wheel can rotate so that point \(P\) ends up at position \(Q\).
- 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.

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

### Solution

### 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.

- \(\frac{1}{3}\) of a full rotation
- \(\frac{1}{2}\) of a full rotation
- \(\frac{2}{3}\) of a full rotation

### Solution

### Problem 6

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

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