Trigonometric function

Complex case

We can also express as a combination of complex exponential functions using Euler’s formula: $e^{ix}=\cos x+i\sin x$ and $e^{-ix}=\cos x-i\sin x$.

$$\sin x=\frac{e^{ix}-e^{-ix}}{2i}=\frac{z-z^{-1}}{2i}$$ $$\cos x=\frac{e^{ix}+e^{-ix}}{2}=\frac{z+z^{-1}}{2}$$

Trigonometric identities still hold for the complex case $z\in \mathbb{C}$, as do differentiation rules.

Note that we can link to hyperbolic functions:

$$\sin z=\sin x\cosh y+i\cos x\sinh y$$ $$\cos z=\cos x\cosh y-i\sin x\sinh y$$

Observe that this comes as a direct result of the compound angle identities and the definition of the hyperbolic functions!

Laplace transform

We can derive the Laplace transform for trigonometric functions using either complex analysis or integration by parts. From Euler’s formula, we have the equivalent:

$$\int e^{i\omega t}{e}^{-st}dt=\int \cos (\omega t)e^{-st}dt+i\int \sin (\omega t)e^{-st}dt$$

So if we take the real or imaginary part of tche right side, we get $\cos$ or $\sin$ multiplied by some exponential function.

$${\int }_0^{\infty }{e}^{st+i\omega t}dt=\frac{s}{s^2+{\omega }^2}+i\frac{\omega }{s^2+{\omega }^2}$$

This is especially useful when computing the Laplace transform of a trigonometric function.

Basics, real case

$\sin$ and $\cos$ are apart (lead or lag) by $\frac{\pi }{2}$ radians.

See also

• Trigonometric identity
• Trigonometric substitution
• Trigonometric integral
• Hyperbolic function
• Sinusoid