Show that ${\left| {{\mathrm{e}}^{\mathrm{i}\theta }} \right|=1}$.[1] Consider the geometric

Question

Show that ${\left| {{\mathrm{e}}^{\mathrm{i}\theta }} \right|=1}$.[1] Consider the geometric series ${1+\frac{1}{3}{{\mathrm{e}}^{\mathrm{i}\theta }}+\frac{1}{9}{{\mathrm{e}}^{2\mathrm{i}\theta }}+…}$ Write down the common ratio, ${z}$, of the series and show that ${\left| z \right|=\frac{1}{3}}$.[2] Find an expression for the sum to infinity of this series.[2] Hence, show that ${\sin \theta +\frac{1}{3}\sin 2\theta +\frac{1}{9}\sin 3\theta +…=\frac{9\sin \theta }{10-6\cos \theta }}$.[8]

Accepted Answer

(a) 0 (b) ${\frac{1}{3}}$ (c) ${\frac{3}{3 - e^{i	heta}}}$ (d) ${\frac{9\sin	heta}{10 - 6\cos	heta}}$

Complex Numbers - Properties of Complex Conjugates

Complex Numbers - Properties of Complex Conjugates

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