Question

Select the correct answer.
Convert sqrt3 + i to polar form.

Answer 1

The polar form of any complex number can be written as

`z = |z| e^(i\arg(z))`

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

If
`z=\sqrt3+i`, then
`z` has modulus

`|z| = √(\left(\sqrt3\right)^2 + 1^2) = \sqrt4 = 2`

and argument

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

Then

`\sqrt3 + i = 2e^(i\frac\pi6) = 2 \left(\cos\left(\frac\pi6\right) + i \sin\left(\frac\pi6\right)\right)`