# 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 \(2\pi\)?

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

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

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

$\sin{\theta} > 0$ 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.

0

$\frac{\pi}{4}$

$\frac{\pi}{2}$

$\pi$

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

### Problem 3

Here are the graphs of two functions.

- Which is the graph of \(y = \cos(\theta)\)? Explain how you know.
- 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.

The function is periodic.

The maximum value is 1.

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

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

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

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.

4.5

5

6

7

7.5

### 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, \(d\).

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