Question

What will be the argument of the complex number $z=1-cosp+isinp$ ?

• A. $\pi -p$
• B. $\pi /4-p/2$
• C. $\pi /2-p/2$
• D. $\pi -p/2$

Answer 1

Answer: (C)

$arg\left(z\right)=ta{n}^{-1}\left(\frac{sinp}{1-cosp}\right)$
$=ta{n}^{-1}\left(\frac{2sinp/2cosp/2}{1-\left(1-2si{n}^{2}p/2\right)}\right)$
$=ta{n}^{-1}\left(\frac{cosp/2}{sinp/2}\right)=ta{n}^{-1}\left(cot\frac{p}{2}\right)$
$=ta{n}^{-1}\left(tan\left(\frac{\pi }{2}-\frac{p}{2}\right)\right)=\frac{\pi }{2}-\frac{p}{2}$