Derive the Triple Angle Formula Complex Analysis

Question

Derive the Triple Angle Formula using the stated proposition:

Answer 1

We can use the de Moivre's formula:

$$e^{i3\phi}=\cos3\phi+i\sin3\phi=(\cos\phi+i\sin\phi)^3=\cos^3\phi-i\sin^3\phi+3i\cos^2\phi\sin\phi-3\sin^2\phi\cos\phi$$

Identifying real and imaginary parts: $$\cos3\phi=\cos^3\phi-3\sin^2\phi\cos\phi$$ $$\sin3\phi=3\cos^2\phi\sin\phi-\sin^3\phi$$