Further Mathematics 9231: Complex numbers Topical Past Paper Questions

(a) Use de Moivre's theorem to show that $$ \sec7\theta=\frac{\sec^{7}\theta}{64-112\sec^{2}\theta+56\sec^{4}\theta-7\sec^{6}\theta}. $$ (b) Hence obtain the roots of the equation $$ x^{7}-14x^{6}+112x^{4}-224x^{2}+128=0 $$ in the form $ \sec q\pi $, where q is rational. [5]

(a) Use de Moivre's theorem to show that $$ \sin5\theta=16\sin^{5}\theta-20\sin^{3}\theta+5\sin\theta. $$ (b) Hence obtain the roots of the equation $$ 32x^{5}-40x^{3}+10x-\sqrt{3}=0 $$ in the form $ \sin q\pi $, where q is a rational number. [4]

(a) By considering the binomial expansion of $ (z+\frac{1}{z})^7 $, where $ z=\cos\theta+\mathrm{i}\sin\theta $, use de Moivre's theorem to show that $$ \cos^{7}\theta=a\cos7\theta+b\cos5\theta+c\cos3\theta+d\cos\theta, $$ where $a$, $b$, $c$ and $d$ are constants to be determine…