In this lesson students practice using trigonometric functions to model the circular motion of a rider on a carousel. The motion of the riders relative to the center of the carousel has a midline of 0, but the amplitude, period, and horizontal translation all need to be interpreted from the context.
Students use the structure of the unit circle to explain why the graphs of the functions defined by \(y=\sin\left(\frac{\pi}{2} + x\right)\) and \(y= \cos(x)\) are actually the same (MP7), noting that a rotation of \(\frac{\pi}{2}\) in the counterclockwise direction takes the horizontal coordinate of a point on the unit circle to the vertical coordinate of the image point on the unit circle.
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.
Lesson overview
 18.1 Warmup: Comparing Bikes (5 minutes)
 18.2 Activity: Around a Carousel (15 minutes)

18.3 Activity: Modeling the Carousel Motion (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 18.4 Cooldown: A different Carousel (5 minutes)
Learning goals:
 Create trigonometric functions to model circular motion given a description.
 Use the relationship between arc length and radian angle measurements to calculate distance traveled around a circle.
Learning goals (student facing):
 Let's use trigonometric functions to model circular motion.
Learning targets (student facing):
 I can represent a circular motion situation using a graph and an equation.
Required materials:
 Scientific calculators
Standards:
 This lesson builds on the standards:CCSS.HSGC.B.5MS.GC.5CCSS.HSFBF.A.1MS.FBF.1MO.G.C.B.4MO.G.C.B.5MO.A2.BF.A.1
 This lesson builds towards the standard:CCSS.HSFTF.B.5MS.FTF.5
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