# Lesson 4

The Unit Circle (Part 2)

### Problem 1

Angle $$ABC$$ measures $$\frac{\pi}{3}$$ radians, and the coordinates of $$C$$ are about $$(0.5,0.87)$$.

1. The measure of angle $$ABD$$ is $$\frac{2\pi}{3}$$ radians. What are the approximate coordinates of $$D$$? Explain how you know.
2. The measure of angle $$ABE$$ is $$\frac{5\pi}{3}$$ radians. What are the approximate coordinates of $$E$$? Explain how you know.

### Problem 2

Give an angle of rotation centered at the origin that sends point $$P$$ to a location whose $$(x,y)$$ coordinates satisfy the given conditions.

1. $$x > 0$$ and $$y < 0$$
2. $$x < 0$$ and $$y > 0$$
3. $$y < 0$$ and $$x < 0$$

### Problem 3

Lin calculates $$0.97^2 + 0.26^2$$ and finds that it is 1.0085.

1. Explain why $$(0.97,0.26)$$ is not on the unit circle.
2. Is $$(0.97,0.26)$$ a good estimate for the coordinates of a point on the unit circle? Explain how you know.

### Problem 4

The $$x$$-coordinate of a point $$P$$ on the unit circle is 0. If point $$P$$ is the result of rotating the point $$(1,0)$$ by $$\theta$$ radians counterclockwise about the origin, what angle could $$\theta$$ represent? Select all that apply.

A:

0

B:

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

C:

$$\pi$$

D:

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

E:

$$2\pi$$

### Problem 5

Here is triangle $$ABC$$. $$BC$$ is shorter than $$AC$$. Which statements are true? Select all that apply.

A:

$$\sin(A) > 1$$

B:

$$\tan(A) < 1$$

C:

$$\cos(A) < 1$$

D:

$$\sin(A) < \sin(B)$$

E:

$$\cos(A) < \cos(B)$$

F:

$$\tan(A) < \tan(B)$$

### Solution

(From Unit 6, Lesson 2.)

### Problem 6

Angle $$POQ$$ measures one radian. The radius of the circle is 1 unit.

1. What is the length of arc $$PQ$$?
2. Explain why the length of arc $$PQ$$ is less than $$\frac{1}{6}$$ of the full circle.

### Solution

(From Unit 6, Lesson 3.)

### Problem 7

Label these points on the unit circle:

1. $$Q$$ is the image of $$P$$ after a $$\frac{11\pi}{6}$$ rotation with center $$O$$.
2. $$R$$ is the image of $$P$$ after a $$\frac{3\pi}{2}$$ rotation with center $$O$$.
3. $$U$$ is the image of $$P$$ after a $$\frac{2\pi}{3}$$ rotation with center $$O$$.
4. $$V$$ is the image of $$P$$ after a $$\frac{\pi}{3}$$ rotation with center $$O$$.