Question

Convert the complex number -5+5√3 i into polar form.
Can someone explain to me the process of doing this? Thanks.

Answer 1

if you have a complex number as
\(\Large x+iy \)

then to convert it into polar form
\(\Large r\angle \theta \)
we use
\(\Large r = \sqrt{x^2+y^2}\)

and \(\Large \theta = tan^{-1}(\dfrac{y}{x})\)

Answer 2

here,
\(\Large x+iy = -5+5\sqrt 3 i\ \\ \Large so, x = -5 , y = 5\sqrt 3 \)

just plug in!

Answer 3

@hartn I don't get it
\[r=\sqrt{x^2+y^2}=\sqrt{(-5)^2+(5\sqrt{3})^2}=10\]
and \(\large tan^- ({\dfrac{y}{x}})= tan^-(\dfrac{5\sqrt{3}}{-5})=tan^-(-\sqrt{3}) \rightarrow \theta =-60 ^0\)
So, I have the polar form is ( \(10, -60^0\))
However, when I test it, x = r cos \(\theta\) = 10 cos (-60) = 5 not -5 . And \( y = r sin\theta = 10*sin (-60) = -5\sqrt{3}\) which are not the given information.
What's wrong, Please explain me.

Answer 4

@hartnn

Answer 5

tan inverse (-sqrt 3)
will have 2 angles as answers
which one would u pick ?

Answer 6

one in 2nd quadrant, one in 4th quadrant

now see the co-ordinates in the question, to which quadrant does they belong ?

Answer 7

r is most definitely 10

Answer 8

Ok, got you!! I forgot the signs of x, y and I used calculator to find tan inverse, so that it gave me -60 degree
Thanks for make it clear.

Answer 9

welcome :)

Answer 10

how does tangent inverse 3 have two angles as answers? All i get is -60 degrees

Answer 11

tan is periodic with 180 degrees.
so each value of tan repeats after 180 degrees
so, in 0 to 360, there will always be 2 angles for any value of tan

Answer 12

if one of them is theta
other will be theta + 180
or theta - 180

Answer 13

now -60 is in 4th quadrant

add 180 to it,
and you will get the angle in 2nd quadrant

Answer 14

we need angle in 2nd quadrant because the the co-ordinates in the question are in 2nd quadrant

Answer 15

ohh alright thanks

Answer 16

welcome ^_^