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

$$\mathrm{P}\left(\mathrm{z}\right)=(1+i\sqrt{3})z^2-(-4+4i)z+2i\cos \left(\frac{\pi }{5}\right)-2\sin \left(\frac{\pi }{5}\right)$$$$\mathrm{Let}S\mathrm{denote}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{roots}\mathrm{of}\mathrm{P}\left(z\right)$$$$\mathrm{a})\mathrm{Express}S\mathrm{in}\mathrm{algebraic}\mathrm{form}\mathrm{then}\mathrm{in}\mathrm{exponential}\mathrm{form}.$$$$\mathrm{b}.\mathrm{Deduce}\mathrm{the}\mathrm{exact}\mathrm{values}\mathrm{of}\cos \left(\frac{5\pi }{12}\right)\mathrm{and}\sin \left(\frac{5\pi }{12}\right).$$

Answered by mindispower last updated on 16/Nov/21

$$S=\frac{-4+4i}{1+i\sqrt{3}}=(-1+i)(1-i\sqrt{3})=2\sqrt{2}.e^{\frac{3\pi }{4}i}.e^{-\frac{i\pi }{3}}$$$$=-1+\sqrt{3}+i(1+\sqrt{3})=2\sqrt{2}.e^{\frac{5\pi }{12}i}$$$$\Rightarrow e^{i\frac{5\pi }{12}}=\frac{\sqrt{3}-1}{2\sqrt{2}}+\frac{i(1+\sqrt{3})}{2\sqrt{2}}$$$$cos\left(\frac{5\pi }{12}\right)=\frac{\sqrt{3}-1}{2\sqrt{2}}=\frac{\sqrt{6}-\sqrt{2}}{4}$$$$sin\left(\frac{5\pi }{12}\right)=\frac{\sqrt{2}+\sqrt{6}}{4}$$

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