This lesson begins by asking students to make the logical step that negative radians mean rotating clockwise since positive radians mean rotating counterclockwise. Returning to the familiar context of a clockface and the rotation of an hour hand, they use the structure of the unit circle to develop strategies for identifying negative radian measurements (MP7). Next, students create visual displays for the graphs of \(y=\cos(\theta)\) and \(y=\sin(\theta)\) for negative and positive angles, highlighting some important features of the graphs and what those features say about the periodic nature of the functions. In later lessons, they will transform cosine and sine functions to model different situations.
This lesson includes an optional activity that offers students more practice working with the graphs of cosine and sine.
Lesson overview
 11.1 Warmup: Rewind to the Windmill (5 minutes)
 11.2 Activity: Math Talk: The Hour Hand (10 minutes)

11.3 Activity: The Big Picture for Cosine and Sine (15 minutes)
 Includes "Are you Ready for More?" extension problem
 11.4 Optional Activity: Cosine and Sine Together (15 minutes)
 Lesson Synthesis
 11.5 Cooldown: Turn it Around Again (5 minutes)
Learning goals:
 Interpret graphs of cosine and sine for input values less than 0.
 Interpret (orally and in writing) the meaning of angle measures less than 0 radians on the unit circle.
Learning goals (student facing):
 Let’s think about the value of cosine and sine for all types of inputs.
Learning targets (student facing):
 I understand how to find the values of cosine and sine for inputs less than 0 radians.
Required materials:
 Tools for creating a visual display
Required Preparation
 Graphing technology is required for the optional activity Cosine and Sine Together.
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