Euler's Identity on the Complex Plane
This animation traces the point e^{iθ} as it moves around the unit circle in the complex plane, showing how the angle θ controls both the real (cosine) and imaginary (sine) coordinates. As θ sweeps from 0 to π, the point rotates halfway around, landing exactly on -1, visually revealing why e^{iπ} + 1 = 0. It helps students connect exponential functions, rotation, and trigonometry in one geometric picture.
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The prompt that made it
Show Euler identity on the complex plane
Make the next one in this series
- Animate e^{iθ} for a full 2π rotation and show sine and cosine as shadows on the axes
- Show how multiplying complex numbers corresponds to adding angles and multiplying lengths
- Derive Euler's formula visually from the Taylor series of sine, cosine, and exponential functions
