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

Answer:

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

Step-by-step explanation:

The complex number given is

$z = (\, \cos(\frac{\pi}{16}) + i\sin(\frac{\pi}{16}) \,)^{1/2}$

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

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