Question

show that (√3/2 +i/2)3=i​

Answer 1

Answer:

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Answer 2

$\displaystyle \sf{ { \bigg( \frac{ \sqrt{3} }{2} + \frac{i}{2} \bigg)}^{3} = i} \: \: \: is \: proved$

Given :

$\displaystyle \sf{ { \bigg( \frac{ \sqrt{3} }{2} + \frac{i}{2} \bigg)}^{3} = i}$

To find :

To prove the expression

Formula :

De Moviers Theorem

$\displaystyle \sf{ { ( cos \theta + i \sin \theta)}^{n} =cos n\theta + i \sin n\theta }$

Solution :

Step 1 of 2 :

Write down the given equation to prove

The given equation to prove is

$\displaystyle \sf{ { \bigg( \frac{ \sqrt{3} }{2} + \frac{i}{2} \bigg)}^{3} = i}$

Step 2 of 2 :

Prove the expression

LHS

$\displaystyle \sf{ = { \bigg( \frac{ \sqrt{3} }{2} + \frac{i}{2} \bigg)}^{3} }$

$\displaystyle \sf{ = { \bigg( \cos \: \frac{\pi}{6} + i \: \sin \: \frac{\pi}{6} \bigg)}^{3} }$

$\displaystyle \sf{ = { \bigg( \cos \: \frac{3\pi}{6} + i \: \sin \: \frac{3\pi}{6} \bigg)}^{} }$

$\displaystyle \sf{ = { \bigg( \cos \: \frac{\pi}{2} + i \: \sin \: \frac{\pi}{2} \bigg)}^{} }$

$\displaystyle \sf{ =0 + i.1 }$

$\displaystyle \sf{ = i}$

= RHS

Hence the proof follows