1. Euler's formula is a fundamental equation in complex analysis that establishes a deep relationship between trigonometric functions and the exponential function. 2. The formula states that for any real number $x$, $$e^{ix} = \cos x + i \sin x$$ where $e$ is the base of the natural logarithm, $i$ is the imaginary unit with $i^2 = -1$, and $\cos$ and $\sin$ are the cosine and sine functions respectively. 3. This formula is extremely useful in fields such as engineering, physics, and mathematics because it allows us to represent complex numbers in exponential form. 4. A special case of Euler's formula is when $x = \pi$, which gives Euler's identity: $$e^{i\pi} + 1 = 0$$ This identity beautifully links five fundamental mathematical constants: 0, 1, $\pi$, $e$, and $i$. 5. To summarize, Euler's rule (or formula) is: $$e^{ix} = \cos x + i \sin x$$