The purpose of this lesson is for students to extend their understanding of cosine and sine from ratios of right triangles or the \(x\)- and \(y\)-coordinates of points on a unit circle for a specific angle to understanding them as functions which have an angle measure as an input. Building on their earlier work studying the structure of the unit circle, students plot points for the graphs of \(y=\cos(\theta)\) and \(y=\sin(\theta)\) for values of \(\theta\) from 0 to \(2\pi\) radians (MP7). Students also use technology to graph several trigonometric functions, including \(y=\cos^2(\theta)+\sin^2(\theta)\), recalling earlier work with the Pythagorean Identity.
In future lessons, students will build on this work to create new functions based on cosine and sine that model periodic motion, such as the height of the blade of a windmill or how the amount of the face of the moon illuminated changes over the course of a lunar month.
- Comprehend cosine and sine as functions whose input is an angle $\theta$ and output is the $x$- or $y$-coordinate of the corresponding point on the unit circle.
- Draw and label the graphs of $y=\cos(\theta)$ and $y=\sin(\theta)$ for values of $\theta$ from 0 to $2\pi$.
- Let’s graph cosine and sine.
Devices are required for the digital version of the activity “Do the Wave.”
- I can use the coordinates of points on the unit circle to graph the cosine and sine functions.
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 \(x\).