# Lesson 3

The Unit Circle (Part 1)

### Problem 1

$$C$$ is a circle with radius $$r$$. Which of the following is true? Select all that apply.

A:

The diameter of $$C$$ is $$2r$$.

B:

The circumference of $$C$$ is $$\pi r$$.

C:

The circumference of $$C$$ is $$2\pi r$$.

D:

One quarter of the circle has length $$\frac{\pi r}{4}$$.

E:

One quarter of the circle has length $$\frac{\pi r}{2}$$.

### Problem 2

angle measure rotation
0 0
$$\frac{\pi}{6}$$
$$\frac{1}{8}$$
$$\frac{1}{6}$$
$$\frac{\pi}{2}$$
$$\frac{2\pi}{3}$$
$$\frac{1}{2}$$
$$\frac{3\pi}{2}$$
$$\frac{7}{8}$$
1

The table shows an angle measure in radians and the amount of rotation about a circle corresponding to the angle. For example, $$2\pi$$ radians corresponds to 1 full rotation. Complete the table.

### Problem 3

A wheel has a radius of 1 foot. After the wheel has traveled a certain distance in the counterclockwise direction, the point $$P$$ has returned to its original position. How many feet could the wheel have traveled? Select all that apply.

A:

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

B:

$$\pi$$

C:

$$2\pi$$

D:

$$5\pi$$

E:

$$10\pi$$

### Problem 4

Here are some points labeled on the unit circle:

1. What is the measure in radians of angle $$POR$$?
2. Angle $$POQ$$ is halfway between 0 radians and angle $$POR$$. What is the measure in radians of angle $$POQ$$?
3. Label point $$U$$ on the circle so that the measure of angle $$POU$$ is $$\frac{3\pi}{4}$$.
4. Label point $$V$$ on the circle so that the measure of angle $$POV$$ is $$\frac{3\pi}{2}$$.

### Problem 5

1. Mark the points on the unit circle with $$x$$-coordinate $$\frac{4}{5}$$.

2. What are the $$y$$-coordinates of those points? Explain how you know.

### Problem 6

The point $$(8, 15)$$ lies on a circle centered at $$(0,0)$$. Where does the circle intersect the $$x$$-axis? Where does the circle intersect the $$y$$-axis? Explain how you know.

### Solution

(From Unit 6, Lesson 1.)

### Problem 7

Triangles $$ABC$$ and $$DEF$$ are similar. Explain why $$\tan(A) = \tan(D)$$.