(1+cos(π/7)+isin(π/7))14= ?how to get answer........................................................

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Algebra> (1+cos( π/7)+isin ( π/7)) 14 = ? how to g...

(1+cos(π/7)+isin(π/7))¹⁴= ?
how to get answer........................................................

milan puzovic , 7 Years ago

Grade 10

1 Answers

Damith Dilhan

We can use $e^{i\theta}$ form for this problem where,

$Cos(\theta)+iSin(\theta) =e^{i\theta}$

Before that we have to remove 1 from equation.In order to do that we substitute $1+Cos(\pi/7)=2cos^{2}(\pi/14)$ and $Sin(\pi/7)=2sin(\pi/14)cos(\pi/14)$ ,

Then,

$(1+cos(\pi /7)+isin(\pi/7))^{14}=(2cos^{2}(\pi/14)+i2sin(\pi/14)cos(\pi/14))^{14}$ $(2cos^{2}(\pi/14)+i2sin(\pi/14)cos(\pi/14))^{14}=2^{14}cos^{14}(\pi/14)(cos(\pi/14)+isin(\pi/14))^{14}$ $2^{14}cos^{14}(\pi/14)(cos(\pi/14)+isin(\pi/14))^{14}=2^{14}cos^{14}(\pi/14)(e^{i\pi/14})^{14}$ $2^{14}cos^{14}(\pi/14)(e^{i\pi/14})^{14}=2^{14}cos^{14}(\pi/14)e^{i\pi}$ $2^{14}cos^{14}(\pi/14)e^{i\pi}=2^{14}cos^{14}(\pi/14)(cos(\pi)+isin(\pi))$ $2^{14}cos^{14}(\pi/14)(cos(\pi)+isin(\pi))=16382.24*(-1)=-16382.24$