# 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.

- \(y = 5\sin(\theta) +10\)
- \(y = 8\sin(\theta) + 20\)
- \(y = 4\sin(\theta) + 15\)

### Problem 2

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

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

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

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

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

### Problem 3

Here is a graph of a trigonometric function.

Which equation does the graph represent?

\(y = 2\sin\left(\theta\right)\)

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

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

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

### 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.

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

### Problem 5

- Explain how to use a unit circle to find a point \(P\) with \(x\)-coordinate \(\cos(\frac{23\pi}{24})\).
- Use a unit circle to estimate the value of \(\cos(\frac{23\pi}{24})\).

### Problem 6

- What are some ways in which the tangent function is similar to sine and cosine?
- What are some ways in which the tangent function is different from sine and cosine?

### Problem 7

Match the trigonometric expressions with their graphs.