# Geometric significance of trigonometric functions for the complex plane

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...

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 $$(a,b)\mapsto a+ib$$.