Interactive trigonometry lesson
How does a circle become a sine wave?
See one idea in two representations: motion around the unit circle and the graph of the point’s vertical coordinate. The live board builds the connection one landmark at a time.
The short answer
A sine graph is the vertical coordinate of a point moving around a circle, recorded against its angle. The circle supplies the range and repetition; the moving point supplies each plotted height.
What happens in the lesson
- 01
Start with motion, not a memorized curve
The lesson begins with a point on a unit circle and a radius marking its angle. The learner can identify the point’s height before a sine graph appears. This makes the graph a record of something observable rather than a shape to copy.
- 02
Match landmark angles to landmark heights
At 0, π/2, π, 3π/2, and 2π, the point’s vertical coordinate is 0, 1, 0, −1, and 0. Those five values become anchor points on the graph. The board keeps the circle and plot visible together so each coordinate has a geometric source.
- 03
Trace the continuous change between landmarks
The curve is not created by connecting unrelated answers. As the point turns continuously, its height changes continuously. Near the top and bottom the vertical change slows; near the horizontal axis it changes more quickly.
- 04
Read amplitude and period from the circle
The radius sets the maximum distance above and below the midline, so it controls amplitude. One full revolution returns the point to its starting state, so the horizontal length of one cycle is 2π radians.
- 05
Check transfer by changing the representation
A useful follow-up asks the learner to predict what changes for a circle of radius 2, clockwise motion, or a different starting point. That tests whether amplitude, direction, and phase are understood as features of the motion.
What this example is designed to teach
- Each horizontal graph position represents an angle, and each vertical position is the circle point’s height.
- Amplitude comes from radius; period comes from the angle required for one full revolution.
- The key points 0, 1, 0, −1, 0 are consequences of the circle, not an arbitrary mnemonic.
- Transformations of sine can be interpreted as changes to the motion, scale, or starting position.
Direct answers
Frequently asked questions
Why does circular motion produce a sine wave?+
For a point at angle θ on the unit circle, the vertical coordinate is sin(θ). As the angle increases at a constant rate, plotting that vertical coordinate against the angle traces the sine graph.
What determines the amplitude?+
The radius determines the largest vertical displacement. A unit circle has radius 1, so its vertical coordinate ranges from −1 to 1 and the resulting sine graph has amplitude 1.
Why is the period 2π?+
After the angle increases by 2π radians, the point completes one revolution and returns to the same vertical coordinate and direction of motion. The same pattern therefore repeats every 2π.