Welcome to the fascinating world of Euler's theorem. This remarkable formula connects the exponential function with trigonometry through complex numbers. On the unit circle, any point can be represented as cosine theta plus i sine theta, which equals e to the power of i theta.
The complex exponential function e to the power of i theta traces out points on the unit circle. As theta increases from zero, the point moves counterclockwise around the circle. The real part gives us cosine theta, while the imaginary part gives us sine theta. This beautiful relationship shows how exponentials and trigonometry are intimately connected.
At special angles, Euler's formula gives us remarkable results. At theta equals zero, e to the i times zero equals one. At pi over two, we get i. Most famously, at theta equals pi, we get negative one, leading to Euler's identity: e to the i pi plus one equals zero. This equation connects five fundamental mathematical constants in one beautiful expression.
Euler's theorem decomposes complex exponentials into sine and cosine waves. This decomposition is fundamental in signal processing, quantum mechanics, and electrical engineering. The real part gives us the cosine wave, while the imaginary part gives us the sine wave. Together, they describe oscillatory phenomena throughout nature and technology.