# Lesson 9

Introduction to Trigonometric Functions

### Problem 1

Which statement is not true for the function $$f$$ given by $$f(\theta) = \sin(\theta)$$, for values of $$\theta$$ between 0 and $$2\pi$$?

A:

The outputs of the function range from -1 to 1.

B:

$$\sin{\theta} = 1$$ only when $$\theta = \frac{\pi}{2}$$

C:

$$\sin{\theta} = 0$$ only when $$\theta = 0$$

D:

$$\sin{\theta} > 0$$ for $$0 < \theta < \pi$$

### Problem 2

Angle $$\theta$$, measured in radians, satisfies $$\cos(\theta) = 0$$. What could the value of $$\theta$$ be? Select all that apply.

A:

0

B:

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

C:

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

D:

$$\pi$$

E:

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

### Problem 3

Here are the graphs of two functions.

1. Which is the graph of $$y = \cos(\theta)$$? Explain how you know.
2. Which is the graph of $$y = \sin(\theta)$$? Explain how you know.

### Problem 4

Which statements are true for both functions $$y = \cos(\theta)$$ and $$y = \sin(\theta)$$? Select all that apply.

A:

The function is periodic.

B:

The maximum value is 1.

C:

The maximum value occurs at $$\theta = 0$$.

D:

The period of the function is $$2\pi$$.

E:

The function has a value of about 0.71 when $$\theta = \frac{\pi}{4}$$.

F:

The function has a value of about 0.71 when $$\theta = \frac{3\pi}{4}$$.

### Problem 5

Here is a graph of a function $$f$$.

The function $$f$$ is either defined by $$f(\theta) = \cos^2(\theta) + \sin^2(\theta)$$ or $$f(\theta) = \cos^2(\theta) - \sin^2(\theta)$$. Which definition is correct? Explain how you know.

### Problem 6

The minute hand on a clock is 1.5 feet long. The end of the minute hand is 6 feet above the ground at one time each hour. How many feet above the ground could the center of the clock be? Select all that apply.

A:

4.5

B:

5

C:

6

D:

7

E:

7.5

### Solution

(From Unit 6, Lesson 7.)

### Problem 7

Here is a graph of the water level height, $$h$$, in feet, relative to a fixed mark, measured at a beach over several days, $$d$$.

1. Explain why the water level is a function of time.
2. Describe how the water level varies each day.
3. What does it mean in this context for the water level to be a periodic function of time?