Unit 7, Lesson 11
Extending the Domain of Trigonometric Functions
Algebra 2
11.1
Warm-up
Rewind to the Windmill
Instructional Routines
Materials
None
Activity 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
Activity
NoneActivity 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.)
Student Lesson Summary
The functions
Notice that between
- 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
- 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.
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