# Thread: Complex numbers: Argument of Z help

1. ## Complex numbers: Argument of Z help

I'm having some problems with finding, in radians, the argument of z and adding rotations of 180 to keep it in range. I know that I have to use:

$tan=\frac{y}{x}$

and I know exact trigonometric ratios, and the ASTC in the polar plane. But I don't know where Arg Z is positive, and I don't know when to add $2\pi$ or even just $\pi$ (pi = 180 in radians), to keep Arg Z in the range of $-\pi < 0 <\pi$. As you can see I am very confused, any help would be great.

2. Consider 1 + i and - 1 - i

In both cases arctan(y/x) = pi/4

since 1 + i is in the first quadr arg(z) = pi/4

hoewever for - 1 -i we know this is in the 3d quadrant so we add pi for the arg

Similarly consider 1 - i and -1 + i

arctan(y/x) = -pi/4 for -1 + i is in the 2 quadrant so add pi to find argz
for 1- i is in the fourth quadrant so add 2pi to find argz

In general locate the quadrant in which z lies to determine arg z

3. Originally Posted by RAz
I'm having some problems with finding, in radians, the argument of z and adding rotations of 180 to keep it in range. I know that I have to use:

$tan=\frac{y}{x}$

and I know exact trigonometric ratios, and the ASTC in the polar plane. But I don't know where Arg Z is positive, and I don't know when to add $2\pi$ or even just $\pi$ (pi = 180 in radians), to keep Arg Z in the range of $-\pi < 0 <\pi$. As you can see I am very confused, any help would be great.
Plot your value of z on an Argand diagram. That makes it very easy to see the ballpark your argument lies in.

4. This function will work.
$
\arg \left( {x + yi} \right) = \left\{ {\begin{array}{ll}
{\arctan \left( {y/x} \right),} & {x > 0} \\
{\arctan \left( {y/x} \right) + \pi ,} & {x < 0\;\& \,y > 0} \\
{\arctan \left( {y/x} \right) - \pi ,} & {x < 0\;\& \,y < 0} \\

\end{array} } \right.$

5. Since you want to keep Arg z between $-\pi$ and $\pi$, Arg z is positive as long as the complex part is postive, negative if the comples part is negative.

6. Originally Posted by RAz
I'm having some problems with finding, in radians, the argument of z and adding rotations of 180 to keep it in range. I know that I have to use:

$tan=\frac{y}{x}$

and I know exact trigonometric ratios, and the ASTC in the polar plane. But I don't know where Arg Z is positive, and I don't know when to add $2\pi$ or even just $\pi$ (pi = 180 in radians), to keep Arg Z in the range of $-\pi < 0 <\pi$. As you can see I am very confused, any help would be great.
First find angle $\alpha$ using $\tan\alpha=\left|\frac{y}{x}\right|$,where $\alpha$ is an angle lying between $0$ and $\frac{\pi}{2}$

Now $arg(z)$ is defined as the angle which the line segment joining origin and $z$ makes with positive direction of the real axis.
Accordingly,
If $x>0,y>0$,then $arg(z)=\alpha$

If $x<0,y>0$,then $arg(z)=\pi-\alpha$

If $x<0,y<0$,then $arg(z)=\alpha-\pi$

If $x>0,y<0$,then $arg(z)=-\alpha$

If $y=0$ and $x>0$,then $arg(z)=0$,If $y=0$ and $x<0$,then $arg(z)=\pi$

If $x=0$ and $y>0$,then $arg(z)=\frac{\pi}{2}$,If $x=0$ and $y<0$,then $arg(z)=-\frac{\pi}{2}$

Draw all these cases on the graph and keep the definition of arg(z)(underlined part) foremost in your mind and you will understand immediately.

7. Originally Posted by Plato
This function will work.
$
\arg \left( {x + yi} \right) = \left\{ {\begin{array}{ll}
{\arctan \left( {y/x} \right),} & {x > 0} \\
{\arctan \left( {y/x} \right) + \pi ,} & {x < 0\;\& \,y > 0} \\
{\arctan \left( {y/x} \right) - \pi ,} & {x < 0\;\& \,y < 0} \\
\end{array} } \right.$
To the former two posters,
With the exception $x=0$ that function answers all of your points.
What problem do you have with that?

8. None whatsoever Plato.

It is just that I remembered the time when I had got stuck in the same concept and finally came out of it when my teacher had explained it to me in the manner I have written.

9. Thanks for your help everyone.

### why argument lies between -180& 180

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