Trigonometric form. De Moivre's theorem

A complex number $z = a + bi$ can also be written using its distance from the origin and the angle it makes with the positive real axis. This is called the trigonometric (or polar) form: $z = r( ext{cos} heta + i ext{sin} heta)$ where $r = |z| = \sqrt{a^2 + b^2}$ is the modulus, and $\theta$ is the argument (angle). De Moivre's theorem states that for any integer $n$, $( ext{cos} heta + i ext{sin} heta)^n = ext{cos}(n heta) + i ext{sin}(n heta)$ This makes it easy to raise complex numbers to powers.

Important properties

• The modulus $r$ is always non-negative.

• The argument $\theta$ is usually measured in radians.

• Multiplying two complex numbers in trigonometric form multiplies their moduli and adds their arguments.

• De Moivre's theorem helps compute powers and roots of complex numbers.