Question

How do you write the complex number in trigonometric form `1+3i`?

Answer 1

`sqrt10(cos(1.249)+isin(1.249))`

Explanation:

To convert from `color(blue)"complex to trigonometric form"`

That is `x+yitor(costheta+isintheta)`

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(r=sqrt(x^2+y^2))color(white)(a/a)|)))`

and `color(red)(bar(ul(|color(white)(a/a)color(black)(theta=tan^-1(y/x))color(white)(a/a)|))) (-pi < theta <= pi)`

here x = 1 and y = 3

`rArrr=sqrt(1^2+3^2)=sqrt10`

Now 1 + 3i is in the 1st quadrant so `theta` must be in the 1st quadrant.

`theta=tan^-1(3)=1.249" rad" larr" in 1st quad."`

`rArr1+3itosqrt10(cos(1.249)+isin(1.249))`