Lesson 11

Extending the Domain of Trigonometric Functions

Lesson Narrative

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 clock face 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.

Learning Goals

Teacher Facing

  • 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.

Student Facing

  • Let’s think about the value of cosine and sine for all types of inputs.

Required Preparation

Graphing technology is required for the optional activity “Cosine and Sine Together.”

Learning Targets

Student Facing

  • I understand how to find the values of cosine and sine for inputs less than 0 radians.

CCSS Standards

Addressing

Glossary Entries

  • periodic function

    A function whose values repeat at regular intervals. If \(f\) is a periodic function then there is a number \(p\), called the period, so that \(f(x + p) = f(x)\) for all inputs \(x\).