Lesson 9

Introduction to Trigonometric Functions

The practice problem answers are available at one of our IM Certified Partners

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.
Graph. 

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 \(f\).

Graph. 

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

(From Algebra2, 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, \(d\).

Coordinate plane, horizontal, d, 0 to 1 point 5 by 1, vertical, h, negative 3 to 3. The graph oscillates between negative 3 and 2 with irregular maximums and minimums. Ask for assistance.
  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?
(From Algebra2, Unit 6, Lesson 8.)