Question

If $z=\cos \frac{\pi }{4}+i\sin \frac{\pi }{6},$ then

• A. $|z|=1,\arg \left(z\right)=\frac{\pi }{4}$
• B. $|z|=1,\arg \left(z\right)=\frac{\pi }{6}$
• C. $|z|=\frac{\sqrt{3}}{2},\arg \left(z\right)=\frac{5\pi }{24}$
• D. $|z|=\frac{\sqrt{3}}{2},\arg \left(z\right)={\tan }^{-1}\frac{1}{\sqrt{2}}$

Answer 1

Answer: (A)

$|z|=1,\arg \left(z\right)=\frac{\pi }{4}$

$z=\cos \frac{\pi }{4}+i\sin \frac{\pi }{6}$

$z=\frac{1}{\sqrt{2}}+i\frac{1}{2}$

$\theta ={\tan }^{-1}\frac{\frac{1}{2}}{\frac{1}{\sqrt{2}}}={\tan }^{-1}\left(\frac{\sqrt{2}}{2}\right)$

$\theta ={\tan }^{-1}\left(\frac{1}{\sqrt{2}}\right)$

$\therefore argz={\tan }^{-1}\frac{1}{\sqrt{2}}$

$|z|=\sqrt{{\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{1}{2}\right)}^2}=\sqrt{\frac{1}{2}+\frac{1}{4}}$

$=\sqrt{\frac{2+1}{4}}$

$=\frac{\sqrt{3}}{2}$

$z=\cos \frac{\pi }{4}+i\sin \frac{pi}{4}$

$argz=angle=\frac{\pi }{4}$

and $|z|={\cos }^2\frac{\pi }{4}+{\sin }^2\frac{\pi }{4}=1$