Work out de moivre theorem instantly with clear inputs, formula shown and shareable results.
De Moivre's theorem says (r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ). It converts a repeated multiplication into a single scaling and rotation, and it is the source of the multiple-angle trigonometric identities.
De Moivre's theorem
[r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)
The modulus becomes 64 and the argument 180°, so the result is −64.
Expanding (cos θ + i sin θ)³ and matching real parts yields cos 3θ = 4cos³θ − 3cos θ immediately.