1.14 Complex roots and De Moivre’s theorem questions for IB Maths AA

Express $z$ in the form $r{e}^{\mathrm{i}\theta }$, where $r>0$ and $-\pi <\theta \le \pi$.

Hence find $z^6$.

Express $z_1$ and $z_2$ in the form $r{\text{e}}^{\text{i}\theta }$, where $r>0$ and $-\pi <\theta \le \pi$.

Hence write down $z_1{z}_2$ in the form $r{\text{e}}^{\text{i}\theta }$.

Find the least positive integer $n$ for which ${\left(z_1{z}_2\right)}^n$ is purely imaginary.

For this value of $n$, find ${\left(z_1{z}_2\right)}^n$.

For $a>0$, explain why $\arg (a+\text{i})=\arctan \left({\displaystyle \frac{1}{a}}\right)$.

Find $(2+\text{i})(3+\text{i})$.

Hence show that $\arctan \left({\displaystyle \frac{1}{2}}\right)+\arctan \left({\displaystyle \frac{1}{3}}\right)={\displaystyle \frac{\pi }{4}}$.

Let $a,b\in \mathbb{R}$, where $a>1$ and $b>1$.

Show that $(a+\text{i})(b+\text{i})=(ab-1)+(a+b)\text{i}$.

Show that $\arctan \left({\displaystyle \frac{1}{a}}\right)+\arctan \left({\displaystyle \frac{1}{b}}\right)={\displaystyle \frac{\pi }{4}}$ if and only if $(a-1)(b-1)=2$.

Hence find all the pairs of integers $a$ and $b$, with $1<a\le b$, such that $\arctan \left({\displaystyle \frac{1}{a}}\right)+\arctan \left({\displaystyle \frac{1}{b}}\right)={\displaystyle \frac{\pi }{4}}$.

Find $(3+\text{i})(7+\text{i})$.

Hence show that $\arctan \left({\displaystyle \frac{1}{3}}\right)+\arctan \left({\displaystyle \frac{1}{7}}\right)=\arctan \left({\displaystyle \frac{1}{2}}\right)$.

Hence show that ${\displaystyle \frac{\pi }{4}}=2\arctan \left({\displaystyle \frac{1}{3}}\right)+\arctan \left({\displaystyle \frac{1}{7}}\right)$.

Show that $(5+\text{i}{)}^4=476+480\text{i}$.

Hence show that $(5+\text{i}{)}^4(239-\text{i})=114244(1+\text{i})$.

Hence show that ${\displaystyle \frac{\pi }{4}}=4\arctan \left({\displaystyle \frac{1}{5}}\right)-\arctan \left({\displaystyle \frac{1}{239}}\right)$.

For small values of $x$, $\arctan x\approx x-{\displaystyle \frac{x^3}{3}}$.

Use this approximation and the result from part (e)(iii) to find an approximate value for $\pi$. Give your answer correct to five significant figures.

Using the same approximation with ${\displaystyle \frac{\pi }{4}}=\arctan 1$ gives $\pi \approx 2.67$. Suggest why the result from part (e)(iii) gives a much better approximation.

Solve the equation, giving your answers in the form $a+b\text{i}$, where $a,b\in \mathbb{R}$.

Let $z_1$ be the root with $\text{Im}\left(z_1\right)>0$ and let $z_2$ be the other root.

Express $z_1$ and $z_2$ in the form $r{\text{e}}^{\text{i}\theta }$, where $r>0$ and $-\pi <\theta \le \pi$.

Find ${\displaystyle \frac{z_1}{z_2}}$, giving your answer both in the form $r{\text{e}}^{\text{i}\theta }$ and in the form $a+b\text{i}$.

Show that $z_1^2=-2z_2$.

On an Argand diagram, the points A, B and C represent $z_1$, $z_2$ and $z_1^2$ respectively.

Find the exact area of triangle ABC.

Show that $z_1^3=z_2^3$, and find their common value.

Find $\mathrm{\mid }w\mathrm{\mid }$ and $\arg w$.

Describe fully the geometric transformation that maps ${\text{P}}_n$ to ${\text{P}}_{n+1}$.

Find $z_3$ in the form $a+b\text{i}$.

Find the smallest value of $n$ for which ${\text{P}}_n$ lies in the fourth quadrant.

Show that $\mid z_{n+1}-z_n\mid =8\sqrt{0.37}\times {0.9}^n$.

A particle travels in straight lines from ${\text{P}}_0$ to ${\text{P}}_1$, then to ${\text{P}}_2$, then to ${\text{P}}_3$, and so on indefinitely.

Find the total distance travelled by the particle.

Find the smallest value of $n$ for which the distance ${\text{OP}}_n$ is less than $1$.

By expanding $(\cos \theta +\mathrm{i}\sin \theta {)}^2$ and using De Moivre's theorem, show that $T_2(x)=2x^2-1$.

By expanding $(\cos \theta +\mathrm{i}\sin \theta {)}^4$, find $T_4(x)$.

Let $z=\cos \theta +\mathrm{i}\sin \theta$. You may use the result $z^n+z^{-n}=2\cos n\theta$.

Show that $\cos (n+1)\theta +\cos (n-1)\theta =2\cos \theta \cos n\theta$, for $n\in {\mathbb{Z}}^{+}$.

Hence write down an expression for $T_{n+1}(x)$ in terms of $T_n(x)$ and $T_{n-1}(x)$, and use it to find $T_5(x)$.

Explain why, for $n\ge 1$, $T_n(x)$ has degree $n$ and leading coefficient $2^{n-1}$.

Use your GDC to find the zeros of $T_5(x)$.

Show that the zeros of $T_n(x)$ are $x=\cos {\displaystyle \frac{(2k-1)\pi }{2n}}$, for $k=1,2,\dots ,n$.

Write down the value of $T_n(1)$.

Let $c_k=\cos {\displaystyle \frac{(2k-1)\pi }{2n}}$. Use parts (e), (g) and (h)(i) to show that $(1-c_1)(1-c_2)\cdots (1-c_n)={\displaystyle \frac{1}{2^{n-1}}}$.

Hence show that ${\displaystyle \prod _{k=1}^n\sin \frac{(2k-1)\pi }{4n}=\frac{\sqrt{2}}{2^n}}$.

Hence find the exact value of $\sin {\displaystyle \frac{\pi }{20}}\sin {\displaystyle \frac{3\pi }{20}}\sin {\displaystyle \frac{5\pi }{20}}\sin {\displaystyle \frac{7\pi }{20}}\sin {\displaystyle \frac{9\pi }{20}}$.