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Propositions from Trigonometric and geometric applications

All propositions from Trigonometric and geometric applications.

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Exponential forms of sine and cosine

For θ∈R\theta\in\mathbb R,

cos⁡θ=eiθ+e−iθ2,sin⁡θ=eiθ−e−iθ2i.\cos\theta=\frac{e^{i\theta}+e^{-i\theta}}2, \qquad \sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i}.

The two identities follow by adding and subtracting Euler's formula and its conjugate form. They let powers of sine and cosine be expanded with complex arithmetic, where powers of eiθe^{i\theta} combine easily: (eiθ)n=einθ(e^{i\theta})^n=e^{in\theta}.

  1. [1]:Polar form
  2. [2]:Expand powers of sine and cosine
  3. [3]:Trigonometric polynomials and a differential operator

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Exponential forms of sine and cosine