AcademyComplex Form
Academy
Complex Trig Formulae
Level 1 - Math I (Physics) topic page in Complex Form.
Complex Trigonometric Formulae
Complex exponentials let us rewrite trigonometric functions as exponential combinations. This is useful in physics because oscillations, rotations, and wave phases often simplify when written with exponentials.
Notation
\(\theta\)
an angle or phase
\(x\)
a real or complex input variable
\(\cosh x\)
hyperbolic cosine of x
\(\sinh x\)
hyperbolic sine of x
Complex Exponential Forms
Euler's formula is the starting point:
Euler's
\[e^{i\theta} = \cos\theta + i\sin\theta\]
Adding the equations for \(e^{i\theta}\) and \(e^{-i\theta}\) isolates cosine. Subtracting them isolates sine:
Cosine from Exponential
\[\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}\]
Sine from Exponential
\[\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}\]
Hyperbolic Connections
The hyperbolic functions are defined by similar exponential combinations, but without the imaginary unit in the exponent:
Hyperbolic Cosine
\[\cosh x = \frac{e^x + e^{-x}}{2}\]
Hyperbolic Sine
\[\sinh x = \frac{e^x - e^{-x}}{2}\]
Useful Identities
Circular Identity
\[\cos^2\theta + \sin^2\theta = 1\]
Hyperbolic Identity
\[\cosh^2 x - \sinh^2 x = 1\]