Different ways of writing complex number arguments

Question

In complex numbers , when the argument of a complex number lies in the second quadrant we usually say that it’s argument is $\pi -\theta$ however what is the meaning of the notation $\theta-\pi$ ? For example , if I wanted to write the modulus argument form of $1+ i\tan\theta$ when $\theta$ lies in the second quadrant I would usually write it as $-\sec\theta \space\text{cis}\space(\pi-\theta)$ however the notation of $-\sec\theta\space\text{cis}(\theta-\pi)$ is also valid . How are both the same ?

Answer 1

$1+i\tan \theta =\sec \theta(\cos \theta + i\sin \theta)=\sec \theta \space \text{cis}\space \theta\\\hspace{43pt}=\sec \theta (-\cos(\pi-\theta)+i\sin(\pi-\theta))\\\hspace{43pt}= -\sec \theta (\cos (\pi-\theta) -i\sin(\pi-\theta))\\\hspace{43pt} =-\sec \theta(\cos(\theta-\pi)+i\sin(\theta-\pi))\\\hspace{43pt} =-\sec\theta \space \text{cis}\space(\theta-\pi)$

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$-\sec \theta \space\text{cis}\space (\pi - \theta) = -\sec \theta(\cos(\pi-\theta)+i\sin(\pi-\theta))\\\hspace{75pt}= -\sec \theta (-\cos \theta+i\sin \theta)\\\hspace{75pt}=\sec\theta(\cos \theta-i\sin\theta)\\\hspace{75pt}=\sec \theta\space\text{cis}\space(-\theta) \ne \sec \theta \space \text{cis}\space \theta\\\hspace{75pt}=1-i\tan\theta \ne 1+i\tan \theta $

So, they're not actually the same.

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As $\theta\in (\frac\pi2,\pi)$,

($\pi-\theta$) means $Q_1$ so imaginary part of $-\sec\theta \space \text{cis}\space(\theta-\pi)$ would lie in $Q_4$.

($\theta-\pi$) means $Q_4$ so imaginary part of $-\sec\theta \space \text{cis}\space(\pi-\theta)$ would lie in $Q_1$.

Also, imaginary part of $1+i\tan \theta$, which is $\tan \theta$, lies in $Q_1$ so that verifies this.