Argand diagram transformations

Let z=r(costheta+isintheta), where r>0, theta in R and n in Z. Then zn = rn[cos(ntheta) + i sin(ntheta)]. For positive integer powers, the identity also holds when r=0. In particular, when r=1, one has (cos theta + i sin theta)n = cos(ntheta) + i sin(ntheta).

Let \(z=r(\cos\theta+i\sin\theta)\), where \(r>0\), \(\theta\in\mathbb R\) and \(n\in\mathbb Z\). Then

\[z^n = r^n[\cos(n\theta) + i \sin(n\theta)].\]

For positive integer powers, the identity also holds when \(r=0\). In particular, when \(r=1\), one has

\[(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta).\]

Properties of Trigonometric and Exponential Forms

Complex Numbers

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

De Moivre's Theorem

De Moivre's Theorem