Hi everybody,
Question,
Find the argument of z for each of the following in the interval [0,2pie]. (Give exact answers where possible)
z=3+2i
the answer is 0.588
please show the working out aswell. Thanks everyone.
complex numbers written in general as
$\displaystyle z=x+iy$ right
then the argument of z given by
$\displaystyle \tan \theta = \frac{y}{x}$ since the argument is $\displaystyle \theta $ values so to find theta values take arctan ( $\displaystyle tan^{-1} $ ) for both sides you will have
$\displaystyle \theta = \tan^{-1} \frac{y}{x} $ the argument of z denoted by arg(z)
note:- argument of z have infinitely many values
the principle argument of z denoted by Arg(z) is the unique value of $\displaystyle \theta $ such that $\displaystyle -\pi < \theta <= \pi $ as you can see the principle argument take one value
you can say that
$\displaystyle arg(z)=Arg(z)+2n\pi$ n integar number
Examples:-
Find the argument and the principle argument to the following:
$\displaystyle 1)z=\sqrt{3}+i $ y=1 and x=sqrt{3} so
$\displaystyle Arg(z)=\tan^{-1} \frac{1}{\sqrt{3}}$
$\displaystyle Arg(z)=\frac{\pi}{6}$
$\displaystyle arg(z)=\frac{\pi}{6}+2n\pi$
$\displaystyle 2)z=1+\sqrt{3}i $
$\displaystyle Arg(z)=\tan^{-1}\frac{\sqrt{3}}{1}$
$\displaystyle Arg(z)=\frac{\pi}{3}$
$\displaystyle arg(z)=\frac{\pi}{3}+2n\pi$
$\displaystyle 3)z=-2+2i$
$\displaystyle Arg(z)=\tan^{-1}\frac{2}{-2}$
$\displaystyle Arg(z)=\tan^{-1}(-1)$
the tan is negative in the second quarter and in the third quarter but the principle argument take the values in the interval $\displaystyle \left(-\pi,\pi\right]$
or to make it easier to determine the angle write it
$\displaystyle Arg(z)=-\tan^{-1}(1)$
so
$\displaystyle Arg(z)=\frac{-\pi}{4}$
$\displaystyle arg(z)=\frac{-\pi}{4}+2n\pi$
$\displaystyle 4)z=-5+7i $
$\displaystyle Arg(z)=\tan^{-1} \frac{7}{-5} $ to make it easier to determine the angle
$\displaystyle Arg(z)=-\tan^{-1} \frac{7}{5} $
$\displaystyle Arg(z)=-54.462 $
$\displaystyle arg(z)=-54.462+2n\pi$
I think it is clear now