Question

simplify (sin π/6+i cos π/6) ^18
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Answer 1

Answer:

it is correct answer

Step-by-step explanation:

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Answer 2

Answer:

Value of ( sin(π/6) +icos(π/6) )¹⁸ is 1 .

Step-by-step explanation:

• Eulerian form of complex number : If z = ( x + iy ) is a complex number, then it can be written as

$z = r {e}^{iθ}$

where, r = | z | and θ = arg ( z )

• This is called Eulerian form and

${e}^{iθ} = cosθ - i \: sinθ$

• De-Moivre's Theorem : If n is an integer, then

${( \cosθ + i \sinθ )}^{n} = \cos(nθ) + i\sin(nθ)$

To find :

• ( sin(π/6) +icos(π/6) )¹⁸

Solution:

• The given equation is not in standard Eulerian form. Converting in standard Eulerian form, we get

${ (\sin( \frac{\pi}{6} ) + i \cos( \frac{\pi}{6} ) ) }^{18} = {(\cos( \frac{\pi}{2} - \frac{\pi}{6} ) + i \sin( \frac{\pi}{2} - \frac{\pi}{6} ) )}^{18} \\ = { (\cos( \frac{\pi}{3} ) - i \sin( \frac{\pi}{3} )) }^{18}$

• From De-Moivre's Theorem, we get;

$\cos( \frac{18\pi}{3} ) + i \sin( \frac{18\pi}{3} ) = \cos(6\pi) + i \sin(6\pi) \\ = 1$

• Hence, value of ( sin(π/6) +icos(π/6) )¹⁸ is 1.