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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...

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

J. W. Tanner
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LBJFS
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  • 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 '19 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 '19 at 16:41