Question

Express 2+6i√3÷3 +i√3 in polar form?

Answer 1

`(10.583*(cos(79.10°)+i*sin(79.10°)))/(3.464*(cos(30°)+i*sin(30°)))`

Explanation:

First let's state the conversion between cartesian form and polar form

`z=a+bi`
`z = r*(cos(theta)+i*sin(theta))`
Where `r = sqrt(a^2+b^2)` and `theta = arctan(b/a)`

Let's break the equation into two smaller equations

The first: `z_1 = 2+i*6sqrt(3)`

`r_1 = sqrt(2^2+(6sqrt(3))^2) = 10.583`
`theta_1 = arctan((6sqrt(3))/2) = 79.10°`

And the second: `z_2 = 3 +i*sqrt(3)`

`r_2 = sqrt(3^2+sqrt(3)^2) = 3.464`
`theta_2 = arctan(sqrt(3)/3) = 30°`

Thus we can make the equation:

`(10.583*(cos(79.10°)+i*sin(79.10°)))/(3.464*(cos(30°)+i*sin(30°)))`