Complex numbers and arguments

Question

Hello, i am a bit confused with when finding the arguments of complex numbers

How do you know whether to add or minus π?

For example;

-√3 - i

arg = tan^-1(1/√3)

= π/6

why do you then go -(π - π/6)

to get -5/6π as the final answer?

Thank you.

Answer 1

The arguments of complex number is 

As in the case of computing ø  , we must be careful in determining the argument  ø of complex number.

From the above argument value tells us the value of  ø  upto the possible addition of π , we must look at the sign of imagenary part (y) .

If  y is negitive  add π to argument , otherwise no .

Let us take an example -√3 - i

Since y = -1 is negative, we have θ = 7π/6

Now verify the result 

Complex number z = r ( cos ø + i sin ø )

z = 2  ( cos ( 7π/6 ) + sin (7π/6) )

z = 2 cos ( 7π/6 ) + 2i sin (7π/6)

z = 2  ( -√3/2 ) + 2i (-1/2)

z = -√3 - i