Unit 6, Lesson 11
Extending the Domain of Trigonometric Functions
Integrated Math 3
11.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this Warm-up, students consider the meaning of negative angles on the unit circle as they make sense of the cosine or sine of a negative angle (MP1). In the following activity, students will use their thinking here to identify specific negative radian angles associated with rotations of the hour hand of a clock.
Launch
If needed, display this image of the windmill from an earlier activity, reminding students that the radius of the circle rotates around is 1 meter:
Activity
NonePriya 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.
Student Response
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Building on Student Thinking
Activity Synthesis
Select students to share their thinking about what Priya means. The important point here is that positive radian measure is a counterclockwise rotation on the unit circle, so we can think of negative radian measure as a clockwise rotation on the unit circle.
Here are some questions for discussion:
- “If point is 1 meter from the center of rotation, what does the value of tell us? ?” (Since is at the lowest point at , the value of tells us that the -coordinate at this position lines up with the -coordinate of the center, and the value of tells us that the -coordinate is 1 meter below the center.)
- “What is another negative angle that would have these same values for the coordinates of ?” ( radians since it is one complete clockwise rotation from radians.)
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.