Question

Find the value of (1-i)^100.

(a) 2^100 (cos⁡100π-isin100π)

(b) 2^100 (cos⁡25π-isin25π)

(c) 2^50 (cos⁡25π-isin25π)

(d) 2^50 (cos⁡100π-isin100π)

I had been asked this question in an online interview.

Enquiry is from DeMoivre’s Theorem in portion Complex Numbers of Engineering Mathematics

Answer 1

The correct choice is (c) 2^50 (cos⁡25π-isin25π)

For explanation I would say: We know that,

\(1-i=\sqrt 2 (\frac{1}{\sqrt 2}-\frac{i}{\sqrt 2})=\sqrt 2 (cos \frac{\pi}{4}-isin \frac{\pi}{4})\)

\((1-i) ^{100}=(\sqrt 2 (cos \frac{\pi}{4}-isin \frac{\pi}{4})) ^{100}=2^{50}((cos \frac{\pi}{4}-isin \frac{\pi}{4})) ^{100}\)

By Applying the DeMoivre’s Theorem

\((1-i)^{100}=2^{50} (cos \frac{100\pi}{4}-isin \frac{100\pi}{4})\)

\((1-i)^{100}=2^{50} (cos⁡25\pi-sin25\pi)\).