Lesson 10

Beyond $2\pi$

Lesson Narrative

The purpose of this lesson is for students to extend their understanding of the cosine and sine functions to inputs greater than \(2\pi\) radians. They first consider what an angle greater than \(2\pi\) radians could mean on a unit circle. Students then return to the context of a windmill and interpret graphs depicting the coordinates of the end of one of the blades for different rotations.

The final activity gives students the opportunity to think in terms of both rotations and radians while developing strategies for working with angles greater than 1 full rotation. A key takeaway here is that all angles greater than \(2\pi\) correspond to an angle between 0 and \(2\pi\), which relates directly back to cosine and sine being periodic functions with period \(2\pi\) (MP8). Students also learn that they can use technology to calculate cosine and sine values for these larger angles directly. In the following lesson, they will extend the domain of these trigonometric functions again to include negative inputs.

Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

Learning Goals

Teacher Facing

  • Interpret graphs of cosine and sine for input values greater than $2\pi$.
  • Interpret (in writing) the meaning of angle measures larger than $2\pi$ radians on the unit circle.

Student Facing

  • Let’s go around a circle more than once.

Learning Targets

Student Facing

  • I understand how to find the values of cosine and sine for inputs greater than $2\pi$ radians.

CCSS Standards

Building On


Building Towards

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 \(x\).