1.13 Polar and Euler form 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.

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

Write down $z^{\ast }$ in the form $r{\text{e}}^{\text{i}\theta }$, where $r>0$ and $-\pi <\theta \le \pi$.

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

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

Hence find ${\displaystyle \frac{z_2}{z_1}}$ in the form $r{\text{e}}^{\text{i}\theta }$.

Use your answer to part (b) to find the area of triangle ${\text{OZ}}_1{\text{Z}}_2$.

By applying integration by parts once to $C$, with $u=\cos bx$, show that $aC-bS=e^{ax}\cos bx$.

By applying integration by parts once to $S$, with $u=\sin bx$, show that $bC+aS=e^{ax}\sin bx$.

Hence show that $C={\displaystyle \frac{e^{ax}(a\cos bx+b\sin bx)}{a^2+b^2}}$ and find a similar expression for $S$.

Use Euler's formula to show that $C+\mathrm{i}S={\displaystyle \int e^{(a+b\mathrm{i})x}\mathrm{d}x}$.

It may be assumed that ${\displaystyle \int e^{kx}\mathrm{d}x=\frac{1}{k}{e}^{kx}}$ also holds for a complex constant $k\mathrm{\ne }0$.

Use this result to obtain $C$ and $S$ again, showing that your answers agree with part (b).

Find the exact value of ${\displaystyle {\int }_0^{\pi }{e}^x\sin 3x\mathrm{d}x}$.

Use de Moivre's theorem to show that $\cos 3x=4{\cos }^{3}x-3\cos x$.

Hence find the exact value of ${\displaystyle {\int }_0^{\frac{\pi }{2}}{e}^x{\cos }^{3}x\mathrm{d}x}$.

Show that ${\displaystyle {\int }_0^{\mathrm{\infty }}{e}^{-x}\cos bx\mathrm{d}x=\frac{1}{1+b^2}}$.