Lesson 13

Amplitude and Midline

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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.
Graph. 

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.

A windmill with 5 blades shaped like a tall trapezoid. On the outside end of one of the blade, in the center, a point labeled P.

​​​​​​

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.

Circle on a x y axis, center O. Point P on the circle on the positive x axis. Point Q on the circle in the fourth quadrant, halfway between the neagive y axis and positive x axis.
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.

A circle with center at the origin of an x y plane with grid.

Solution

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(From Unit 6, Lesson 12.)