argument of z.

**johnsy123** · February 7th 2010, 11:42 PM

the given example in the book states the argument of z=4+4i.

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

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

**Prove It** · February 7th 2010, 11:49 PM

Originally Posted by johnsy123

the given example in the book states the argument of z=4+4i.

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

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

It's actually

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

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

$\tan{\theta} = 1$

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

$\theta = 45^\circ$ .

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

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

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

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

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