Question

Find the trigonometric form of the number 3 + √3i.

Answer 1

Any complex number
`z` can be written in trigonometric form as

`z = |z| e^(i\arg(z)) = |z| \left(\cos(\arg(z)) + i \sin(\arg(z))\right)`

where
`|z|` is the modulus of
`z` and
`\arg(z)` is its argument, i.e. the angle
`z` makes with the positive real axis in the complex plane.

We have

`|z| = √(3^2 + \left(\sqrt3\right)^2) = √(12) = 2\sqrt3`

and

`\arg(z) = \tan^(-1)\left(\frac{\sqrt3}3\right) = \tan^(-1)\left(\frac1{\sqrt3}\right) = \frac\pi6`

Then

`3 + \sqrt3 \, i = \boxed{2\sqrt 3 \left(\cos\left(\frac\pi6\right) + i \sin\left(\frac\pi6\right)\right)}`