Exponential Form & de Moivre's Theorem (Edexcel A Level Further Maths: Core Pure): Exam Questions

In an Argand diagram, the points $A$, $B$ and $C$ are the vertices of an equilateral triangle with its centre at the origin. The point $A$ represents the complex number $6+2\mathrm{i}$.

Find the complex numbers represented by the points $B$ and $C$, giving your answers in the form $x+\mathrm{i}y$, where $x$ and $y$ are real and exact.

The points $D$, $E$ and $F$ are the midpoints of the sides of triangle $ABC$.

Find the exact area of triangle $DEF$.

Use de Moivre’s theorem to prove that

$\sin 7\theta =7\sin \theta -56{\sin }^3\theta +112{\sin }^5\theta -64{\sin }^7\theta$

Hence find the distinct roots of the equation

$1+7x-56x^3+112x^5-64x^7=0$

giving your answer to 3 decimal places where appropriate.

A complex number $z$ has modulus 1 and argument $\theta$.

Show that

$z^n+\frac{1}{z^n}=2\cos n\theta ,$      $n\in {\mathrm{\mathbb{Z}}}^{+}$

Hence, show that

${\cos }^4\theta =\frac{1}{8}(\cos 4\theta +4\cos 2\theta +3)$

The complex number $z={\text{e}}^{\text{i}\theta }$, where $\theta$ is real.

Show that

$z^n-\frac{1}{z^n}\equiv 2\text{i} \text{sin} n\theta$

where $n$ is a positive integer.

Show that

${\text{sin}}^5\theta =\frac{1}{16}(\text{sin}5\theta -5\text{sin}3\theta +10\text{sin}\theta )$

Hence, making your reasoning clear, determine all the solutions of

$\text{sin}5\theta -5\text{sin}3\theta +14\text{sin}\theta =0$

in the interval $0\le \theta <2\pi$.

Determine the roots of the equation $z^4=1$, giving your answers in the form ${\text{e}}^{\text{i}\theta }$ where $0\le \theta <2\pi$.

Show the roots of the equation in part (a) on a single Argand diagram.

Show that $(1+\text{i}{)}^4=-4$.

Hence, or otherwise, solve the equation $z^4+4=0$, giving your answers in the form $r{\text{e}}^{\text{i}\theta }$ where $0\le \theta <2\pi$.