Question

What is the trigonometric form of `(2-25i)`?

Answer 1

`sqrt629(cos(1.49)-sin(1.49))`

Explanation:

The trigonometric form of a complex number `z=x+iy`

is `color(red)(bar(ul(|color(white)(2/2)color(black)(z=r(costheta+isintheta))color(white)(2/2)|)))`

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

`color(red)(bar(ul(|color(white)(2/2)color(black)(theta=tan^-1(y/x))color(white)(2/2)|)))`
where `-pi < theta <= pi`

Here `x=2" and " y=-25`

`rArrr=sqrt(2^2+(-25)^2)=sqrt629`

Now 2 - 25i is in the 4th quadrant, so we must ensure that `theta` is in the 4th quadrant.

`rArrtheta=tan^-1(-25/2)=-1.49larr" in 4th quadrant"`

`rArr2-25i=sqrt629(cos(-1.49)+isin(-1.49))`

which can also be expressed as.

`2-25i=sqrt629(cos(1.49)-isin(1.49))`