If α=cos(118π)+i sin(118π), then Re(α+ α2+ α3+ α4+ α5) is

Question

If α=cos(118π​)+i sin(118π​), then Re(α+ α2+ α3+ α4+ α5) is

Filo · Accepted Answer

α=cosA+isinA=cisA,  where A=118π ​  z=α+α 2+α 3+α 4+α 5=α(α−1α 5−1​ )  ⇒z=cisA(cisA−1(cisA) 5−1​ )=cisA(1−cosA−isinA 1−cos5A−isin5A ​ )  ⇒z=sin2A​  cisAsin25A​  ​(sin2A​−icos2A​  sin25A​−icos25A​  ​ )=sin2A​  cisAsin25A​  ​(isin2A​+cos2A​  isin25A​+cos25A​  ​ )  ⇒z=sin2A​  cisAsin25A​ cis2A​=sin2A​  cis3Asin25A​  ​  ⇒Re(z)=sin2A​  cos3Asin25A​  ​  ⇒Re(z)=2sin4A​cos4A​   cos3Asin25A​  ​=2sin(112π ​ )cos(112π ​ )   cos(1124π ​ ) sin(1120π ​ )  ​=−2sin(2π−112π ​ )cos(2π+112π ​ )   cos(1124π ​ ) sin(1120π ​ )  ​  ⇒Re(z)=−2sin(1120π ​ )cos(1124π ​ )   cos(1124π ​ ) sin(1120π ​ )  ​=2−1​ Hence, option B is correct.

Question Text	If $α = cos (\frac{8 π}{11}) + i s in (\frac{8 π}{11})$ , then $R e (α + α^{2} + α^{3} + α^{4} + α^{5})$ is
Answer Type	Text solution:1
Upvotes	150

If $α = cos (\frac{8 π}{11}) + i s in (\frac{8 π}{11})$ , then $R e (α + α^{2} + α^{3} + α^{4} + α^{5})$ is

Text solutionVerified

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If α=cos(118π​)+i sin(118π​), then Re(α+ α2+ α3+ α4+ α5) is

Text solutionVerified

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If $α = cos (\frac{8 π}{11}) + i s in (\frac{8 π}{11})$ , then $R e (α + α^{2} + α^{3} + α^{4} + α^{5})$ is