Question

How do you write `2+3i` in trigonometric form?

Answer 1

To convert, `a+bitor(cos(theta)+isin(theta))`, use:
`r=sqrt(a^2+b^2)`
`theta=tan^-1(b/a)+0,pi,pi,2pi`
Depending on whether the signs of "a" and "b" indicate the, 1st, 2nd, 3rd or 4th quadrant.

Explanation:

Given: `2+3i`

`r = sqrt(2^2+3^2)`

`r = sqrt(4+9)`

`r = sqrt(13)`

The signs of "a" and "b" indicate the 1st quadrant:

`theta = tan^-1(b/a)+0`

`theta = tan^-1(3/2)`

`theta ~~ 56.3^@ or 0.983" radians"`

`2+3i to sqrt(13)(cos(56.3^@)+isin(56.3^@))`