Lesson 14

Transforming Trigonometric Functions

  • Let’s make lots of changes to the graphs of trigonometric functions.

Problem 1

These equations model the vertical position, in feet above the ground, of a point at the end of a windmill blade. For each function, indicate the height of the windmill and the length of the windmill blades.

  1. \(y = 5\sin(\theta) +10\)
  2. \(y = 8\sin(\theta) + 20\)
  3. \(y = 4\sin(\theta) + 15\)

Problem 2

Which expression takes the same value as \(\cos(\theta)\) when \(\theta = 0, \frac{\pi}{2}, \pi,\) and \(\frac{3\pi}{2}\)?

A:

\(\sin\left(\theta -\frac{\pi}{2}\right)\)

B:

\(\sin\left(\theta + \frac{\pi}{2}\right)\)

C:

\(\sin(\theta+\pi)\)

D:

\(\sin(\theta-\pi)\)

Problem 3

Here is a graph of a trigonometric function.

Which equation does the graph represent?

graph of cosine function. amplitude = 2. shifted by the fraction pi over 4 to the right. 
A:

\(y =  2\sin\left(\theta\right)\)

B:

\(y = 2\cos\left(\theta+\frac{\pi}{4}\right)\)

C:

\(y = 2\sin\left(\theta-\frac{\pi}{4}\right)\)

D:

\(y = 2\cos\left(\theta-\frac{\pi}{4}\right)\)

Problem 4

The vertical position \(v\) of a point at the tip of a windmill blade, in feet, is given by \(v(\theta) = 11 + 2\sin\left(\theta+\frac{\pi}{2}\right)\). Here \(\theta\) is the angle of rotation.

  1. How long is the windmill blade? Explain how you know.
  2. What is the height of the windmill? Explain how you know.
  3. Where is the point \(P\) when \(\theta = 0\)?

Problem 5

  1. Explain how to use a unit circle to find a point \(P\) with \(x\)-coordinate \(\cos(\frac{23\pi}{24})\).
  2. Use a unit circle to estimate the value of \(\cos(\frac{23\pi}{24})\).
(From Unit 6, Lesson 5.)

Problem 6

  1. What are some ways in which the tangent function is similar to sine and cosine?
  2. What are some ways in which the tangent function is different from sine and cosine?
(From Unit 6, Lesson 12.)

Problem 7

Match the trigonometric expressions with their graphs.

Graph 1

sine function that begins at 0 comma -3. maximum at the fraction pi over 2 comma -1. minimum at the fraction 3 pi over 2 comma -5. 

Graph 2

cosine function that begins at 0 comma 1. maximum at 2 pi comma 1. minimum at pi comma -5. 

Graph 3

sine function that begins at 0 comma -2. maximum at the fraction pi over 2 comma 1. minimum at the fraction 3 pi over 2 comma -5. 

Graph 4

cosine function that begins at 0 comma -1. maximum at 2 pi comma -1. minimum at pi comma -5. 
A:

\(3\cos(\theta) - 2\)

B:

\(2\cos(\theta) - 3\)

C:

\(3\sin(\theta) -2\)

D:

\(2\sin(\theta)-3\)

1:

Graph 1

2:

Graph 2

3:

Graph 3

4:

Graph 4

(From Unit 6, Lesson 13.)