Further Pure Mathematics WFM01: COMPLEX NUMBERS Topical Past Paper Questions

Given that $$ \sin\theta=\frac{1}{2\mathrm{i}}\Big(\mathrm{e}^{\mathrm{i}\theta}-\mathrm{e}^{-\mathrm{i}\theta}\Big) $$ (a) show that $$ \sin^{4}\theta=\frac{1}{8}\left(A\cos4\theta+B\cos2\theta+C\right) $$ where $A$, $B$ and $C$ are integers to be determined. (4) Using $\cos\the…

(a) Use de Moivre’s theorem to show that $$ \sin5\theta\equiv a\sin^{5}\theta+b\sin^{3}\theta+c\sin\theta $$ where $a$, $b$ and $c$ are integers to be determined. (5) (b) Hence determine the possible exact values of $ \sin^{2}\left(\frac{k\pi}{5}\right) $ where $ k \in Z $ (4)

The transformation T from the z-plane to the w-plane is given by $$ w=\frac{z-\mathrm{i}}{z+1}\quad z\neq-1 $$ Given that T maps the imaginary axis in the z-plane to the circle C in the w-plane, determine (i) the coordinates of the centre of C (ii) the radius of C (7)