# argument of z.

• Feb 7th 2010, 11:42 PM
johnsy123
argument of z.
the given example in the book states the argument of z=4+4i.

1. it has said to use $\displaystyle tan=\frac{4}{4}=1$
2. then it saids that the $\displaystyle arg(z)=\frac{\pi}{4}$

-my concern is, how did it get $\displaystyle arg(z)=\frac{\pi}{4}$ ?
• Feb 7th 2010, 11:49 PM
Prove It
Quote:

Originally Posted by johnsy123
the given example in the book states the argument of z=4+4i.

1. it has said to use $\displaystyle tan=\frac{4}{4}=1$
2. then it saids that the $\displaystyle arg(z)=\frac{\pi}{4}$

-my concern is, how did it get $\displaystyle arg(z)=\frac{\pi}{4}$ ?

It's actually

$\displaystyle \tan{\theta} = \frac{y}{x}$ (since the angle is in the first quadrant)

$\displaystyle \tan{\theta} = \frac{4}{4}$

$\displaystyle \tan{\theta} = 1$

$\displaystyle \theta = \arctan{1}$ (or $\displaystyle \tan^{-1}{1}$, depending on which notation you use)

$\displaystyle \theta = 45^\circ$.

Now since $\displaystyle 360^\circ = 2\pi^C$

$\displaystyle \frac{360^\circ}{8} = \frac{2\pi^C}{8}$

$\displaystyle 45^\circ = \frac{\pi}{4}^C$.

So $\displaystyle \arg{z} = \frac{\pi}{4}^C + 2\pi^Cn$, where $\displaystyle n$ is an integer representing the number of times you have gone around the unit circle.