I have been able to find proofs for trigonometric identities (like the cosine triple angle formula) using De Moivre's theorem (Using De Moivre's Theorem to prove $\cos(3\theta) = 4\cos^3(\theta) – 3\cos(\theta)$ trig identity). This being said, I'm having trouble understanding how a theorem with imaginary parts could "speak" for the reals as well? It's sort of a conceptual question but any math or pictures to clarify would be appreciated…
Geometric significance of trigonometric functions for the complex plane
abstract-algebracomplex numberstrigonometry
Best Answer
If understand your question, you are asking why results for complex numbers apply to real numbers. It follow by the fact that you can identify $\mathbb{R}$ with the real axis of the complex plane $\mathbb{C}$ via the map $x\mapsto x+i0$.
About the De Moivre's formula, it uses the polar coordinates and the angle $\theta$ is a real number which measures the angle with the $x$-axis in $\mathbb{R}^2$, where $\mathbb{R}^2$ is identified with $\mathbb{C}$ using the map $(a,b)\mapsto a+ib$.