(a) Use de Moivre's theorem to show that
\(\cos 5 \theta=16 \cos ^{5} \theta-20 \cos ^{3} \theta+5 \cos \theta\)
(b) Hence obtain the roots of the equation
\(32 x^{5}-40 x^{3}+10 x-\sqrt{2}=0\)
in the form \(\cos (q \pi)\), where \(q\) is a rational number.
Find the roots of the equation \((z+5 \mathrm{i})^{3}=4+4 \sqrt{3} \mathrm{i}\), giving your answers in the form \(r \cos \theta+\mathrm{i}(r \sin \theta-5)\), where \(r\gt 0\) and \(0\lt \theta\lt 2 \pi\).
(a) Use de Moivre's theorem to show that
\(\operatorname{cosec} 7 \theta=\frac{\operatorname{cosec}^{7} \theta}{7 \operatorname{cosec}^{6} \theta-56 \operatorname{cosec}^{4} \theta+112 \operatorname{cosec}^{2} \theta-64}\)
(b) Hence obtain the roots of the equation
\(x^{7}-14 x^{6}+112 x^{4}-224 x^{2}+128=0\)
in the form \(\operatorname{cosec} q \pi\), where \(q\) is rational.
(a) Use de Moivre's theorem to show that
\(\operatorname{cosec} 5 \theta=\frac{\operatorname{cosec}^{5} \theta}{5 \operatorname{cosec}^{4} \theta-20 \operatorname{cosec}^{2} \theta+16} .\)
(b) Hence obtain the roots of the equation
\(x^{5}-10 x^{4}+40 x^{2}-32=0\)
in the form \(\operatorname{cosec}(q \pi)\), where \(q\) is rational.
(a) Use de Moivre's theorem to show that \(\sin ^{6} \theta=-\frac{1}{32}(\cos 6 \theta-6 \cos 4 \theta+15 \cos 2 \theta-10)\).
It is given that \(\cos ^{6} \theta=\frac{1}{32}(\cos 6 \theta+6 \cos 4 \theta+15 \cos 2 \theta+10)\).
(b) Find the exact value of \(\int_{0}^{\frac{1}{3} \pi}\left(\cos ^{6}\left(\frac{1}{4} x\right)+\sin ^{6}\left(\frac{1}{4} x\right)\right) \mathrm{d} x\).
(c) Express each root of the equation \(16 c^{6}+16\left(1-c^{2}\right)^{3}-13=0\) in the form \(\cos k \pi\), where \(k\) is a rational number.
By considering the binomial expansions of \(\left(z+\frac{1}{z}\right)^{5}\) and \(\left(z-\frac{1}{z}\right)^{5}\), where \(z=\cos \theta+\mathrm{i} \sin \theta\), use de Moivre's theorem to show that
\[\tan ^{5} \theta=\frac{\sin 5 \theta-a \sin 3 \theta+b \sin \theta}{\cos 5 \theta+a \cos 3 \theta+b \cos \theta}\]
where \(a\) and \(b\) are integers to be determined.
(i) Let \(z=\cos \theta+\mathrm{i} \sin \theta\). Show that \(z-\frac{1}{z}=2 \mathrm{i} \sin \theta\) and hence express \(16 \sin ^{5} \theta\) in the form \(\sin 5 \theta+p \sin 3 \theta+q \sin \theta\), where \(p\) and \(q\) are integers to be determined.
(ii) Hence find the exact value of \(\int_{0}^{\frac{1}{3} \pi} 16 \sin ^{5} \theta \mathrm{~d} \theta\).
(i) Use de Moivre's theorem to show that
\(\sin 5 \theta=5 \sin \theta-20 \sin ^{3} \theta+16 \sin ^{5} \theta .\)
(ii) Hence explain why the roots of the equation \(16 x^{4}-20 x^{2}+5=0\) are \(x= \pm \sin \frac{1}{5} \pi\) and \(x= \pm \sin \frac{2}{5} \pi\).
(iii) Without using a calculator, find the exact values of
\(\sin \frac{1}{5} \pi \sin \frac{2}{5} \pi \sin \frac{3}{5} \pi \sin \frac{4}{5} \pi \quad \text { and } \quad \sin ^{2}\left(\frac{1}{5} \pi\right)+\sin ^{2}\left(\frac{2}{5} \pi\right) .\)
Use de Moivre's theorem to show that
\(\cos 5 \theta \equiv \cos \theta\left(16 \sin ^{4} \theta-12 \sin ^{2} \theta+1\right)\)
By considering the equation \(\cos 5 \theta=0\), show that the exact value of \(\sin ^{2}\left(\frac{1}{10} \pi\right)\) is \(\frac{3-\sqrt{ } 5}{8}\).
Use de Moivre's theorem to show that
\(\cos 5 \theta \equiv \cos \theta\left(16 \sin ^{4} \theta-12 \sin ^{2} \theta+1\right)\)
By considering the equation \(\cos 5 \theta=0\), show that the exact value of \(\sin ^{2}\left(\frac{1}{10} \pi\right)\) is \(\frac{3-\sqrt{ } 5}{8}\).
3 (i) Write down the fifth roots of unity.
(ii) Find all the roots of the equation
\(z^{10}+z^{5}+1=0\)
giving each root in the form \(\mathrm{e}^{\mathrm{i} \theta}\).
(a) Find \(a\) and \(b\) such that
\[z^{8}-\mathrm{i} z^{5}-z^{3}+\mathrm{i}=\left(z^{5}-a\right)\left(z^{3}-b\right) .\]
(b) Hence find the roots of
\[z^{8}-\mathrm{i} z^{5}-z^{3}+\mathrm{i}=0\]
giving your answers in the form \(r \mathrm{e}^{\mathrm{i} \theta}\), where \(r>0\) and \(0 \leqslant \theta<2 \pi\).
Find all the roots of the equation \((w+1)^{6}=1\), giving your answers in the form \(x+\mathrm{i} y\) where \(x\) and \(y\) are real and exact.
(a) Write down all the roots of the equation \(x^{5}-1=0\).
(b) Use de Moivre's theorem to show that \(\cos 4 \theta=8 \cos ^{4} \theta-8 \cos ^{2} \theta+1\).
(c) Use the results of parts (a) and (b) to express each real root of the equation
\[8 x^{9}-8 x^{7}+x^{5}-8 x^{4}+8 x^{2}-1=0\]
in the form \(\cos k \pi\), where \(k\) is a rational number.
(i) Write down the five fifth roots of unity.
(ii) Hence find all the roots of the equation
\(z^{5}+16+(16 \sqrt{ } 3) i=0\)
giving answers in the form \(r \mathrm{e}^{\mathrm{i} q \pi}\), where \(r\gt 0\) and \(q\) is a rational number. Show these roots on an Argand diagram.
Let \(w\) be a root of the equation in part (ii).
(iii) Show that
\(\sum_{k=0}^{4}\left(\frac{w}{2}\right)^{k}=\frac{3+i \sqrt{ } 3}{2-w} .\)
(iv) Identify the root for which \(|2-w|\) is least.
(a) State the sum of the series \(1+z+z^{2}+\ldots+z^{n-1}\), for \(z \neq 1\).
(b) By letting \(z=\cos \theta+\mathrm{i} \sin \theta\), where \(\cos \theta \neq 1\), show that
\(1+\cos \theta+\cos 2 \theta+\ldots+\cos (n-1) \theta=\frac{1}{2}\left(1-\cos n \theta+\frac{\sin n \theta \sin \theta}{1-\cos \theta}\right) .\)
The diagram shows the curve with equation \(y=\cos x\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).
(c) By considering the sum of the areas of these rectangles, show that
\(\int_{0}^{1} \cos x \mathrm{~d} x\lt \frac{1}{2 n}\left(1-\cos 1+\frac{\sin 1 \sin \frac{1}{n}}{1-\cos \frac{1}{n}}\right)\)
(d) Use a similar method to find, in terms of \(n\), a lower bound for \(\int_{0}^{1} \cos x \mathrm{~d} x\).
(a) State the sum of the series \(1+w+w^{2}+w^{3}+\ldots+w^{n-1}\), for \(w \neq 1\).
(b) Show that \((1+\mathrm{i} \tan \theta)^{k}=\sec ^{k} \theta(\cos k \theta+\mathrm{i} \sin k \theta)\), where \(\theta\) is not an integer multiple of \(\frac{1}{2} \pi\).
(c) By considering \(\sum_{k=0}^{n-1}(1+\mathrm{i} \tan \theta)^{k}\), show that
\(\sum_{k=0}^{n-1} \sec ^{k} \theta \sin k \theta=\cot \theta\left(1-\sec ^{n} \theta \cos n \theta\right),\)
provided \(\theta\) is not an integer multiple of \(\frac{1}{2} \pi\).
(d) Hence find \(\sum_{k=0}^{6 m-1} 2^{k} \sin \left(\frac{1}{3} k \pi\right)\) in terms of \(m\).
(a) State the sum of the series \(z+z^{2}+z^{3}+\ldots+z^{n}\), for \(z \neq 1\).
(b) Given that \(z\) is an \(n\)th root of unity and \(z \neq 1\), deduce that \(1+z+z^{2}+\ldots+z^{n-1}=0\).
(c) Given instead that \(z=\frac{1}{3}(\cos \theta+\mathrm{i} \sin \theta)\), use de Moivre's theorem to show that
\(\sum_{m=1}^{\infty} 3^{-m} \cos m \theta=\frac{3 \cos \theta-1}{10-6 \cos \theta}\)
(a) Show that \(\sum_{r=1}^{n} z^{2 r}=\frac{z^{2 n+1}-z}{z-z^{-1}}\), for \(z \neq 0,1,-1\).
(b) By letting \(z=\cos \theta+\mathrm{i} \sin \theta\), show that, if \(\sin \theta \neq 0\),
\[1+2 \sum_{r=1}^{n} \cos (2 r \theta)=\frac{\sin (2 n+1) \theta}{\sin \theta}\]
State the sum of the series \(z+z^{2}+z^{3}+\ldots+z^{n}\), for \(z \neq 1\).
By letting \(z=\cos \theta+\mathrm{i} \sin \theta\), show that
\(\cos \theta+\cos 2 \theta+\cos 3 \theta+\ldots+\cos n \theta=\frac{\sin \frac{1}{2} n \theta}{\sin \frac{1}{2} \theta} \cos \frac{1}{2}(n+1) \theta\)
where \(\sin \frac{1}{2} \theta \neq 0\).