# Lesson 13

Amplitude and Midline

### Problem 1

For each trigonometric function, indicate the amplitude and midline.

1. $$y = 2\sin(\theta)$$
2. $$y = \cos(\theta) - 5$$
3. $$y= 1.4 \sin(\theta) + 3.5$$

### Solution

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### Problem 2

Here is a graph of the equation $$y = 2\sin(\theta) - 3$$.

1. Indicate the midline on the graph.
2. Use the graph to find the amplitude of this sine equation.

### Solution

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### Problem 3

Select all trigonometric functions with an amplitude of 3.

A:

$$y = 3\sin(\theta) -1$$

B:

$$y = \sin(\theta) + 3$$

C:

$$y = 3\cos(\theta) + 2$$

D:

$$y = \cos(\theta) - 3$$

E:

$$y = 3\sin(\theta)$$

F:

$$y = \cos(\theta - 3)$$

### Solution

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### Problem 4

The center of a windmill is 20 feet off the ground and the blades are 10 feet long.

​​​​​​

rotation angle
of windmill
vertical position
of $$P$$ in feet
$$\frac{\pi}{6}$$
$$\frac{\pi}{3}$$
$$\frac{\pi}{2}$$
$$\pi$$
$$\frac{3\pi}{2}$$
1. Fill out the table showing the vertical position of $$P$$ after the windmill has rotated through the given angle.

2. Write an equation for the function $$f$$ that describes the relationship between the angle of rotation $$\theta$$ and the vertical position of the point $$P$$, $$f(\theta)$$, in feet.

### Solution

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### Problem 5

The measure of angle $$\theta$$, in radians, satisfies $$\sin(\theta) < 0$$. If $$\theta$$ is between 0 and $$2\pi$$ what can you say about the measure of $$\theta$$?

### Solution

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(From Unit 6, Lesson 9.)

### Problem 6

Which rotations, with center $$O$$, take $$P$$ to $$Q$$? Select all that apply.

A:

$$\frac{3\pi}{4}$$ radians

B:

$$\frac{15\pi}{4}$$ radians

C:

$$\frac{7\pi}{4}$$ radians

D:

$$\frac{11\pi}{4}$$ radians

E:

$$\frac{23\pi}{4}$$ radians

### Solution

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(From Unit 6, Lesson 10.)

### Problem 7

The picture shows two points $$P$$ and $$Q$$ on the unit circle.

Explain why the tangent of $$P$$ and $$Q$$ is 2.

### Solution

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(From Unit 6, Lesson 12.)