How might we help students connect the unit-circle representation of trigonometric functions with the graphs of these same functions? Below (and here) is a Web Sketchpad model that gives students the tools to construct the graphs of trigonometric functions by using the unit circle as the driving engine. To get started, just follow the bulleted steps.

You can watch a video that illustrates the construction steps here. And you might also be interested in this webinar video that demonstrates how to model periodic phenomena with desktop Sketchpad.

Use the Point tool to place a point on the orange segment.

Use the Abscissa tool to measure the point’s x-value ( x_{A} ).

Use the Rotate tool to rotate the unit pointâ€”the point at (1, 0)â€”about the origin by an angle equal to your measured x_{A} in radians. Label this rotated point P.

Use the Animate tool to create a button that will animate point A along its domain.

How does point P behave when you drag or animate A along its domain?

If you make a graph of the height of P as a function of A, what will the graph look like? Draw a picture to show your guess about the shape of the graph.

Measure the height of point P using the Ordinate tool.

Use the Plot as (x, y) tool to plot the height y_{P} as a function of the x-value x_{A} of your animated point. Use the Trace widget to trace this plotted point, and then animate point A to trace out the graph. How well does it match your guess?

Your traced graph shows how P moves up and down as you animate A. But P also moves left and right. If you make a graph of x_{P} as a function of x_{A}, what would the graph look like? Draw a picture to show your guess about the shape of this graph.

Use the Abscissa tool to measure x_{P}, plot this new value as a function of x_{A}, and trace the result using a different color. Then animate to see what the new graph looks like.