Unit Circle Waves

This simulation shows a point at angle theta on the unit circle. As the angle sweeps, the point draws a helix, its shadow on the back wall is the sine wave and its shadow on the floor is the cosine wave. Students use the angle slider or Auto-sweep, show or hide the helix, and display the tangent as a length on the line x equals 1. Readouts give the angle and the values of sin, cos and tan. Teachers can use it to link the circle definition to the graphs of trig functions.

• Drag Angle to 30, 45 and 60 degrees and write down the sin and cos readouts.
• Turn on Auto-sweep and watch the wave shadows build as the point turns.
• Switch Helix off and on to separate the circle from the wave graphs.
• Show the tangent and find what happens near 90 degrees.
• Check that sin squared plus cos squared stays at 1 for every angle.

1. 1Relate the coordinates of a point on the unit circle to sine and cosine.
2. 2Describe the period of the sine and cosine graphs.
3. 3Explain why the tangent is undefined at 90 and 270 degrees.
4. 4Verify the identity sin squared plus cos squared equals 1.

Questions teachers ask