# Lesson 11

Extending the Domain of Trigonometric Functions

• Let’s think about the value of cosine and sine for all types of inputs.

### Problem 1

For which of these angles is the sine negative? Select all that apply.

A:

$$\text-\frac{\pi}{4}$$

B:

$$\text-\frac{\pi}{3}$$

C:

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

D:

$$\text-\frac{4\pi}{3}$$

E:

$$\text-\frac{11\pi}{6}$$

### Problem 2

Which of the following are true? Select all that apply.

A:

In the next hour, the minute hand moves through an angle of $$2\pi$$ radians.

B:

In the next 5 minutes, the minute hand will move through an angle of $$\text-\frac{\pi}{6}$$ radians.

C:

After the minute hand moves through an angle of $$\text-\pi$$ radians, it is 3:30 p.m.

D:

When the hour hand moves through an angle of $$\text-\frac{\pi}{6}$$ radians, it is 4:00 p.m.

E:

The angle the minute hand moves through is 12 times the angle the hour hand moves through.

### Problem 3

Plot each point on the unit circle.

1. $$A=(\cos(\text-\frac{\pi}{4}), \sin(\text-\frac{\pi}{4}))$$
2. $$B=(\cos(2\pi),\sin(2\pi))$$
3. $$C=\left(\cos(\frac{16\pi}{3}), \sin(\frac{16\pi}{3})\right)$$
4. $$D=\left(\cos(\text-\frac{16\pi}{3}), \sin(\text-\frac{16\pi}{3})\right)$$

### Problem 4

Which of these statements are true about the function $$f$$ given by $$f(\theta) = \sin(\theta)$$? Select all that apply.

A:

The graph of $$f$$ meets the $$\theta$$-axis at $$0, \pm \pi, \pm 2\pi, \pm 3\pi, \ldots$$

B:

The value of $$f$$ always stays the same when $$\pi$$ radians is added to the input.

C:

The value of $$f$$ always stays the same when $$2\pi$$ radians is added to the input.

D:

The value of $$f$$ always stays the same when $$\text-2\pi$$ radians is added to the input.

E:

The graph of $$f$$ has a maximum when $$\theta = \frac{5\pi}{2}$$ radians.

### Problem 5

Here is a unit circle with a point $$P$$ at $$(1,0)$$.

For each positive angle of rotation of the unit circle around its center listed, indicate on the unit circle where $$P$$ is taken, and give a negative angle of rotation which takes $$P$$ to the same location.

1. $$A$$, $$\frac{\pi}{4}$$ radians
2. $$B$$, $$\frac{\pi}{2}$$ radians
3. $$C$$, $$\pi$$ radians
4. $$D$$, $$\frac{3\pi}{2}$$ radians

### Problem 6

In which quadrant are both the sine and the tangent negative?

A:

first

B:

second

C:

third

D:

fourth

(From Unit 6, Lesson 6.)

### Problem 7

Technology required. Each equation defines a function. Graph each of them to identify which are periodic. Select all that are.

A:

$$y = \sin(\theta)$$

B:

$$y = e^x$$

C:

$$y = x^2 - 2x + 5$$

D:

$$y = \cos(\theta)$$

E:

$$y = 3$$

(From Unit 6, Lesson 8.)

### Problem 8

1. List three different counterclockwise angles of rotation around the center of the circle that take $$P$$ to $$Q$$.
2. Which quadrant(s) are the angles $$\frac{13\pi}{4}$$ and $$\frac{10\pi}{3}$$ radians in? Is the sine of these angles positive or negative?