Tour 6: A Sine Wave Tracer

This tour introduces you to Sketchpad animation as you work through an activity with The Geometer's Sketchpad. In this exploration, you'll construct an animation "engine" that traces out a special curve called a sine wave. Variations of sine curves are the graphs of functions called periodic functions, functions that repeat themselves. The motion of a pendulum and ocean tides are examples of periodic functions.

Sketch and Investigate

  1. Construct a horizontal segment AB.

     

  2. Construct a circle with center A and radius endpoint C.

  3. Construct point D on AB.

  4. Construct a line perpendicular to AB through point D.

    5. Select point D and AB; then, in the Construct menu, choose Perpendicular Line.

  5. Construct point E on the circle.

  6. Construct a line parallel to AB through point E.

  7. Construct point F, the point of intersection of the vertical line through point D and the horizontal line through point E.

Q1 Drag point D and describe what happens to point F.

Q2 Drag point E around the circle and describe what point F does.

Q3 In a minute, you'll create an animation in your sketch that combines these two motions. But first try to guess what the path of point F will be when point D moves to the right along the segment at the same time that point E is moving around the circle. Sketch the path you imagine below.

  1. Make an action button that animates point D forward along AB and point E forward around the circle.

    2. Select points D and E and choose Edit |Action Buttons |Animation. Choose forward in the Direction pop-up menu for point D.

  2. Move point D so that it's just to the right of the circle.

  3. Select point F; then, in the Display menu, choose Trace Intersection.

  4. Press the Animation button.

Q4 In the space below, sketch the path traced by point F. Does the actual path resemble your guess in Q3? How is it different?

  1. Select the circle; then, in the Graph menu, choose Define Unit Circle. You should get a graph with the origin at point A. Point B should lie on the x-axis. The y-coordinate of point F above AB is the value of the sine of <EAD.

Q5 If the circle has a radius of 1 grid unit, what is its circumference in grid units? (Calculate this yourself; don't use Sketchpad to measure it because Sketchpad will measure in inches or centimeters, not grid units.)

  1. Measure the coordinates of point B.

  2. Adjust the segment and the circle until you can make the curve trace back on itself instead of drawing a new curve every time. (Keep point B on the x-axis.)

Q6 What's the relationship between the x-coordinate of point B and the circumference of the circle (in grid units)? Explain why you think this is so.

Extension: "Special Points"

In the next section, you'll add the right triangle associated with point E and also add a new type of button -- Movement buttons -- that will move point E to exact locations corresponding to angles of 30, 45, and 60 degrees.

  1. Measure the coordinates of point E.

  2. Construct the line through point E, perpendicular to the x-axis.

  3. Construct the intersection point of this new line and the x-axis. Label the point G

  1. Construct the point at (1, 0) and label it H. (You can either choose Plot Points from the Graph menu or simply click on the spot to get the point of intersection.)

  2. Measure <HAE.

  3. Hide the perpendicular line and construct segments from A to E, from G to E, and from A to G. Make these three segments thick and red. This is your "reference triangle."

Now you'll create points on the circle at 30, 45, and 60 degrees. You'll do this by rotating point H about the origin, or A.

  1. Double-click point A. This marks it as a center of rotation.

  2. Select point H and choose Rotate from the Transform menu. Type 30 and click Rotate.

  3. Select point E and the new point in that order and choose Movement from the Edit | Action Buttons submenu. Leave the speed at medium. On the Label panel, change the label of the button to "30 degrees."

  4. Hide the new point (at 30 degrees), then press your button. Point E should gracefully move around the circle until <HAE is 30 degrees.

  5. Repeat steps 22 and 23 for 45 and 60 degrees.

Further Challenges