Lesson 13
Amplitude and Midline
 Let's transform the graphs of trigonometric functions
Problem 1
For each trigonometric function, indicate the amplitude and midline.
 \(y = 2\sin(\theta)\)
 \(y = \cos(\theta)  5\)
 \(y= 1.4 \sin(\theta) + 3.5\)
Problem 2
Here is a graph of the equation \(y = 2\sin(\theta)  3\).
 Indicate the midline on the graph.
 Use the graph to find the amplitude of this sine equation.
Problem 3
Select all trigonometric functions with an amplitude of 3.
\(y = 3\sin(\theta) 1\)
\(y = \sin(\theta) + 3\)
\(y = 3\cos(\theta) + 2\)
\(y = \cos(\theta)  3\)
\(y = 3\sin(\theta)\)
\(y = \cos(\theta  3)\)
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 

\(\frac{\pi}{6}\)  
\(\frac{\pi}{3}\)  
\(\frac{\pi}{2}\)  
\(\pi\)  
\(\frac{3\pi}{2}\) 

Fill out the table showing the vertical position of \(P\) after the windmill has rotated through the given angle.

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.
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 \(\theta\)?
Problem 6
Which rotations, with center \(O\), take \(P\) to \(Q\)? Select all that apply.
\(\frac{3\pi}{4}\) radians
\(\frac{15\pi}{4}\) radians
\(\frac{7\pi}{4}\) radians
\(\frac{11\pi}{4}\) radians
\(\frac{23\pi}{4}\) radians
Problem 7
The picture shows two points \(P\) and \(Q\) on the unit circle.
Explain why the tangent of \(P\) and \(Q\) is 2.