How does circular motion generate a sine wave?

As a point rotates around a unit circle at a constant speed, its vertical height (y-coordinate) rises and falls. Plotting this height over time or angle sweeps out a continuous sine wave.

What does frequency do to a wave?

Frequency dictates the number of complete wave cycles that occur within a given interval. Increasing the frequency makes the wave cycles compress horizontally.

Phase shift is a horizontal displacement of the wave left or right along the axis, corresponding to a starting angle offset in circular rotation.

SINE Wave Plotter - Math Simulation

Variable Adjuster

Amplitude (A)2

0.54

Frequency (B)1

0.54

Phase Shift (C)0°

0360

Auto-Sweep Engine

Target Param:

Sweep Speed: 2x

Sine & Cosine Waves

WAVE

Sinusoidal waves demonstrate how circular motion transitions into a linear coordinate wave over time. A point rotating at a constant rate maps its vertical (sine) or horizontal (cosine) displacement directly to the wave curve.

y = A \sin(B\theta + C)

Whiteboard Solver Steps

Step 1

Wave Equation Setup

A sinusoidal wave is characterized by its mathematical function: - $A$ is the Amplitude (vertical scaling / height of peaks): current $A = 2$ . - $B$ is the Frequency multiplier (how squished or stretched the wave cycles are): current $B = 1$ . - $C$ is the Phase shift (horizontal displacement / left-right shift): current $C = 0^\circ = 0.0000\text{ rad}$ .

y = A \sin(B\theta + C) \quad \text{or} \quad y = A \cos(B\theta + C)

Step 2

Wave Period & Wavelength Calculation

The period ( $T$ ) is the horizontal span required to complete one full cycle of the wave. Since a base cycle of sine/cosine spans $360^\circ$ ( $2\pi$ radians), a frequency of $B = 1$ means the wave completes a full cycle in exactly $360.0^\circ$ .

T = \frac{360^\circ}{B} = \frac{360^\circ}{1} = 360.0^\circ \quad (\text{or } \frac{2\pi}{B} \text{ rad})

Step 3

Evaluate Wave Equation

By feeding angular input values ( $\theta$ ) into the trigonometric function, we calculate the vertical displacement ( $y$ ) at each stage, plotting the continuous wave curve across the grid.

y(\theta) = 2 \times \text{sine}(1 \cdot \theta_{rad} + 0.0000)

RV Anveshana