# 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...

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

• thanks, I was struggling to put to words the geometric link between the use of an angle theta (trig functions) and the complex plane as well – Arnold Joseph Feb 20 at 16:01
• Thanks for accepting my answer! I hope you have solved your problems. It is also possible to upvote if it was satisfactory ;-) – LBJFS Feb 20 at 16:41