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Question Number 159322 by Ar Brandon last updated on 15/Nov/21

P (z) = (1 + i \sqrt{3}) z^{2} - (- 4 + 4 i) z + 2 i \cos (\frac{π}{5}) - 2 \sin (\frac{π}{5})

Let S denote the sum of roots of P (z)

a) Express S in algebraic form then in exponential form .

b . Deduce the exact values of \cos (\frac{5 π}{12}) and \sin (\frac{5 π}{12}) .

Answered by mindispower last updated on 16/Nov/21

S = \frac{- 4 + 4 i}{1 + i \sqrt{3}} = (- 1 + i) (1 - i \sqrt{3}) = 2 \sqrt{2} . e^{\frac{3 π}{4} i} . e^{- \frac{i π}{3}}

= - 1 + \sqrt{3} + i (1 + \sqrt{3}) = 2 \sqrt{2} . e^{\frac{5 π}{12} i}

\Rightarrow e^{i \frac{5 π}{12}} = \frac{\sqrt{3} - 1}{2 \sqrt{2}} + \frac{i (1 + \sqrt{3})}{2 \sqrt{2}}

c o s (\frac{5 π}{12}) = \frac{\sqrt{3} - 1}{2 \sqrt{2}} = \frac{\sqrt{6} - \sqrt{2}}{4}

s i n (\frac{5 π}{12}) = \frac{\sqrt{2} + \sqrt{6}}{4}