Priya is thinking about the windmill that has a point, , at at the end of the blade that starts at 0 radians pointing directly to the right.
Right now, is at . Where will be if the windmill rotates radians in the clockwise direction?
Priya says that if the blade rotates radians from its starting position, then will be at the lowest point in its circle of rotation. What do you think Priya means by rotating radians? Do you agree with Priya? Be prepared to explain your reasoning.
11.2
Activity
Waterwheel Height
The blades of a waterwheel are 1 meter long and are centered at , with a point at
Complete the table for the position of point on the waterwheel at each angle.
How do these heights compare to the heights for angles rotated the same amount in the opposite direction?
angle (radians)
height (meters)
0
0
11.3
Activity
The Big Picture for Cosine and Sine
Create a visual display for the following functions. Include a graph of the function from at least to radians, the maximum and minimum value of the function, and the period of the function. Label any intersections that the graph of the function has with the axes.
The -axis is a line of symmetry for one of the two graphs. Which one? Explain how you know.
11.4
Activity
Cosine and Sine Together
Use graphing technology to graph the functions and on the same axes.
Identify two points where the graphs intersect—one with a negative -coordinate, and one with a positive -coordinate. What is the exact -coordinate for each point? Explain or show how you know.
What are the -coordinates of the points of intersection? Explain or show how you know.
What could be the value of , if ? Explain your reasoning.
Student Lesson Summary
The functions and are both periodic, meaning their values repeat at regular intervals. Since the period of both cosine and sine is , the values of these functions repeat any time the input is changed by a multiple of . We can see this in the graph of shown here.
Notice that between and , and then between 0 and , the same wave pattern repeats. Both positive and negative values for can be thought of through the lens of the unit circle, with positive values indicating counterclockwise rotation and negative values indicating clockwise rotation. This means that different important features of the graph occur at regular intervals:
The -intercepts of the graph are at all integer multiples of .
The relative maximums are at and any integer multiple of from there.
The relative minimums are at and any integer multiple of from there.
A graph. Horizontal axis, theta, scale negative 2 pi to 2 pi by pi over 2. Vertical axis, y, negative 1 to 1. There are 3 tick marks between negative 1 and 0 on the vertical axis. There are 3 tick marks between 0 and 1 on the vertical axis. A curve passes through the points negative 2 pi comma 0, negative 3 pi over 2 comma 1, negative pi comma 0, negative pi over 2 comma negative 1, 0 comma 0, pi over 2 comma 1, pi comma 0, 3 pi over 2 comma negative 1, 2 pi comma 0.
The graph of is also periodic, repeating every time the input changes by a multiple of .
The -intercepts are at and any integer multiple of from there.
The relative maximums are at 0 and any integer multiple of from there.
The relative minimums are at and any integer multiple of from there.
A graph. Horizontal axis, theta, scale negative 2 pi to 2 pi by pi over 2. Vertical axis, y, negative 1 to 1. There are 3 tick marks between negative 1 and 0 on the vertical axis. There are 3 tick marks between 0 and 1 on the vertical axis. A curve passes through the points negative 2 pi comma 1, negative 3 pi over 2 comma 0, negative pi comma negative 1, negative pi over 2 comma 0, 0 comma 1, pi over 2 comma 0, pi comma negative 1, 3 pi over 2 comma 0, 2 pi comma 1.
