# 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.

A:

$$\frac{3\pi}{2}$$

B:

$$\frac{\pi}{2}$$

C:

$$\frac{\pi}{4}$$

D:

$$\frac{5\pi}{2}$$

E:

$$\frac{5\pi}{4}$$

### 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 $$R$$.

### Problem 3

Here is a wheel with radius 1 foot.

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?

### 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.

### 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.

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

(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 $$\sin(\theta) = \text-\frac{1}{2}$$?