Euler's formula
/ OY-ler /
Euler's formula is the single equation e^(i theta) = cos theta + i sin theta. In words: raising the constant e to an imaginary power traces out the unit circle — as theta runs from 0 upward, e^(i theta) walks counterclockwise around the circle of radius 1, its real part being cos theta and its imaginary part being sin theta. It is the bridge that ties exponentials (the language of growth and decay) to trigonometry (the language of rotation and waves), and the reason that bridge exists is hidden in the matching Taylor series.
Where does it come from? Expand e^x, cos x, and sin x as their infinite power series, then substitute x = i theta into e^x. The powers of i cycle through 1, i, -1, -i, sorting the terms into two groups: the terms with no leftover i reassemble exactly the series for cos theta, and the terms carrying one i reassemble exactly i times the series for sin theta. The identity falls out. This is also what makes the definition e^(x + i y) = e^x (cos y + i sin y) the natural one for the complex exponential.
Setting theta = pi gives the famous e^(i pi) + 1 = 0, often called the most beautiful equation in mathematics for tying together e, i, pi, 1, and 0. But the real payoff is practical: Euler's formula turns trigonometric identities into easy exponential algebra (cos theta = (e^(i theta) + e^(-i theta))/2, sin theta = (e^(i theta) - e^(-i theta))/(2i)), underlies the exponential form of complex numbers, and is the engine behind Fourier analysis, signal processing, and quantum mechanics.
To prove cos 2 theta = cos^2 theta - sin^2 theta, square Euler: (e^(i theta))^2 = e^(i 2 theta) = cos 2 theta + i sin 2 theta. But (cos theta + i sin theta)^2 = (cos^2 theta - sin^2 theta) + i (2 sin theta cos theta). Matching real parts gives the identity instantly.
Euler's formula turns trig identities into one-line exponential algebra.
e^(i theta) always has modulus 1 (it lives on the unit circle) — so |e^(i theta)| = 1 for every real theta. Do not confuse this with the general e^z, whose modulus is e^(Re z) and can be anything positive.