Find the Argument

**johnsy123** · February 24th 2011, 03:07 AM

Find the Argument of the following.

$Arg\left[cis\left(2pi/3\right)cis\left(pi/4\right)/cis\left(-pi/6\right)cis\left(3pi/4\right)\right]$

**Prove It** · February 24th 2011, 03:14 AM

Do you mean...

$\displaystyle \textrm{Arg}\,{\left[\frac{\textrm{cis}\,{\left(\frac{2\pi}{3}\right)\, \textrm{cis}\,{\left(\frac{\pi}{4}\right)}}}{\text rm{cis}\,{\left(-\frac{\pi}{6}\right)}\,\textrm{cis}\,{\left(\frac{ 3\pi}{4}\right)}}\right]}$ ?

If so, simplify the stuff in the big square brackets first.

**DrSteve** · February 24th 2011, 03:17 AM

When multiplying you add the angles. When dividing you subtract the angles. Finally if your angle is not between $-\pi$ and $\pi$ , add or subtract a multiple of $2\pi$ to get an angle that is.

**pickslides** · February 24th 2011, 03:18 AM

Maybe start by simplifying the term inside the argument using

$\displaystyle z_1\times z_2 = r_1\times r_2 (\cos(\theta_1+\theta_2 ) +i \sin(\theta_1+\theta_2 ) )$

and

$\displaystyle \frac{z_1}{z_2} = \frac{r_1}{ r_2} (\cos(\theta_1-\theta_2 ) +i \sin(\theta_1-\theta_2 ) )$

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