Question

find the modulus and argument of the complex no z= 1+cos2pai/3+ i sin 2 pai/3 ​

Answer 1

Step-by-step explanation:

We have,

$z = 1 + \cos \bigg( \frac{2\pi}{3} \bigg) + i \sin \bigg( \frac{2\pi}{3} \bigg)$

$\implies \: z = 2\cos^{2} \bigg( \frac{\pi}{3} \bigg) + i .2\sin \bigg( \frac{\pi}{3} \bigg) \cos \bigg( \frac{\pi}{3} \bigg)$

$\implies \: z = 2. \bigg( \frac{1}{2} \bigg)^{2} + i .2\bigg( \frac{ \sqrt{3} }{2} \bigg).\bigg( \frac{1}{2} \bigg)$

$\implies \: z = 2. \bigg( \frac{1}{4} \bigg) + i .2\bigg( \frac{ \sqrt{3} }{4} \bigg)$

$\implies \: z = \bigg( \frac{1}{2} \bigg) + i \bigg( \frac{ \sqrt{3} }{2} \bigg)$

$\implies \: z = \frac{1}{2} (1 + i \sqrt{3} )$

So,

$|z| = \sqrt{ \bigg( \frac{1}{2} \bigg)^{2} +\bigg( \frac{ \sqrt{3} }{2} \bigg)^{2} }$

$\implies |z| = \sqrt{ \frac{1}{4} + \frac{ 3 }{4} }$

$\implies |z| = \sqrt{ \frac{1 + 3 }{4} }$

$\implies |z| = \sqrt{ \frac{4}{4} }$

$\implies |z| = 1$

And,

$\arg(z) = \tan^{ - 1} \bigg| \frac{ \frac{ \sqrt{3} }{2} }{ \frac{1}{2} } \bigg|$

$\implies\arg(z) = \tan^{ - 1} \bigg| \frac{ \ \sqrt{3}}{ 1 } \bigg|$

$\implies\arg(z) = \tan^{ - 1} | \sqrt{3} |$

$\implies\arg(z) = \tan^{ - 1} \sqrt{3}$

$\implies\arg(z) = \frac{\pi}{3}$