Semester 5 / Calculus / Complex Functions / Trigonometric Functions

Trigonometric Functions

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For any complex number zz,

sin⁡z=∑n=0∞(−1)nz2n+1(2n+1)!,cos⁡z=∑n=0∞(−1)nz2n(2n)!\sin z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n+1}}{(2n+1)!}, \qquad \cos z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n}}{(2n)!}

Exponential Form

eiz=cos⁡z+isin⁡ze^{iz} = \cos z + i\sin z for each z∈Cz \in \mathbb{C}, so

sin⁡z=eiz−e−iz2i,cos⁡z=eiz+e−iz2\sin z = \frac{e^{iz} - e^{-iz}}{2i}, \qquad \cos z = \frac{e^{iz} + e^{-iz}}{2}

Properties

  • sin⁡z\sin z and cos⁡z\cos z are entire functions.
  • Standard real trigonometric identities hold for complex arguments.
  • sin⁡z=sin⁡xcosh⁡y+icos⁡xsinh⁡y\sin z = \sin x \cosh y + i\cos x \sinh y, where z=x+iyz = x + iy
  • cos⁡z=cos⁡xcosh⁡y−isin⁡xsinh⁡y\cos z = \cos x \cosh y - i\sin x \sinh y, where z=x+iyz = x + iy

Other Trigonometric Functions

tan⁡z\tan z, cot⁡z\cot z, sec⁡z\sec z, and csc⁡z\csc z are defined from sin⁡z\sin z and cos⁡z\cos z as in the real case.

Written by September 13, 2026 1 min read
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