Euler's Formula

Euler’s formula says that multiplying by a complex exponential rotates a point in the complex plane by an angle θ. It works because the power series for e^x, cos x, and sin x match up term-by-term when x is replaced by iθ, so the real parts become cos θ and the imaginary parts become sin θ. This gives a compact bridge between exponential growth/decay, rotation, and wave behavior.

Use this when converting between rectangular and polar forms of complex numbers, especially in advanced algebra, signals, and differential equations. It is also useful whenever a rotating vector, oscillation, or sinusoidal motion can be simplified using complex notation.

• Forgetting that θ must be in radians, not degrees
• Dropping the imaginary unit and treating e^(iθ) like a real exponential
• Mixing up the signs of sin θ or using the wrong angle values from the unit circle