Question

Use​ DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [one half (cosine StartFraction pi Over 16 EndFraction plus i sine StartFraction pi Over 16 EndFraction )]Superscript 8

Answer 1

`(\, \cos((\pi)/(16)) + i\sin((\pi)/(16)) \,)^(1/2) = \cos((\pi)/(32)) + i\sin((\pi)/(32)) = 0.99 + i0.09`

Explanation:

The complex number given is

`z = (\, \cos((\pi)/(16)) + i\sin((\pi)/(16)) \,)^(1/2)`

Now, remember that the DeMoivre's theorem states that

`( \cos(x) + i\sin(x) )^n = \cos(nx) + i\sin(nx)`

Then for this case we have that

`(\, \cos((\pi)/(16)) + i\sin((\pi)/(16)) \,)^(1/2) = \cos((\pi)/(32)) + i\sin((\pi)/(32)) = 0.99 + i0.09`