\(e^{i\theta} = \cos(\theta) + i \sin(\theta)\)
\(e^{-i\theta} = \cos(\theta) - i \sin(\theta)\)
Trigonometric Functions in Exponential Form
\(\cos(\theta) = \frac{1}{2} (e^{i\theta} + e^{-i\theta})\)
\(\sin(\theta) = \frac{1}{2}i (e^{i\theta} - e^{-i\theta})\)
\(\cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2}\)
\(\sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2}i\)
\(e^{i\theta} = \cos(\theta) + i \sin(\theta)\)
\(e^{i\theta} = \frac{1}{2} (e^{i\theta} + e^{-i\theta}) + i \sin(\theta) = \frac{1}{2}i (e^{i\theta} - e^{-i\theta})\)
Real and Imaginary Parts
\(\cos x = \operatorname{Re}(e^{ix}) = \frac{e^{ix} + e^{-ix}}{2}\)
\(\sin x = \operatorname{Im}(e^{ix}) = \frac{e^{ix} - e^{-ix}}{2}i\)
\(e^{i\theta} = \cos(\theta) + i \sin(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2} + i\cdot\frac{e^{i\theta} - e^{-i\theta}}{2i}\)
This elegant relationship connects exponential functions with trigonometric functions through complex numbers.