Trace the Sine Function

1 Explore

In this activity, you can use the Animate tool to create and then animate a variable point along the orange domain segment. Label this point x, and then use the Measure Abscissa tool to measure its x-value.

You can then rotate the unit point—the point at (1, 0)—about the origin by an angle equal to your measured x_x in radians. Label this rotated point P.

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

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

Measure the height of P using the Measure Ordinate tool, and then plot the height y_P as a function of the x-value x_x of your animated point. Use one of the Trace tools to trace this plotted point, and then animate point x 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 x. But P also moves left and right.

✏️ If you make a graph of x_P as a function of x_x, what would the graph look like? Draw a picture to show your guess about the shape of this graph.

Use the Measure Abscissa tool to measure x_P, plot this new value as a function of x_x, and trace the result using a different color. Then animate to see what the new graph looks like.