EU02S-OPG

$\nonumber \newcommand{\bevisslut}{$\blacksquare$} \newenvironment{matr}[1]{\hspace{-.8mm}\begin{bmatrix}\hspace{-1mm}\begin{array}{#1}}{\end{array}\hspace{-1mm}\end{bmatrix}\hspace{-.8mm}} \newcommand{\transp}{\hspace{-.6mm}^{\top}} \newcommand{\maengde}[2]{\left\lbrace \hspace{-1mm} \begin{array}{c|c} #1 & #2 \end{array} \hspace{-1mm} \right\rbrace} \newenvironment{eqnalign}[1]{\begin{equation}\begin{array}{#1}}{\end{array}\end{equation}} \newcommand{\eqnl}{} \newcommand{\matind}[3]{{_\mathrm{#1}\mathbf{#2}_\mathrm{#3}}} \newcommand{\vekind}[2]{{_\mathrm{#1}\mathbf{#2}}} \newcommand{\jac}[2]{{\mathrm{Jacobi}_\mathbf{#1} (#2)}} \newcommand{\diver}[2]{{\mathrm{div}\mathbf{#1} (#2)}} \newcommand{\rot}[1]{{\mathbf{rot}\mathbf{(#1)}}} \newcommand{\am}{\mathrm{am}} \newcommand{\gm}{\mathrm{gm}} \newcommand{\E}{\mathrm{E}} \newcommand{\Span}{\mathrm{span}} \newcommand{\mU}{\mathbf{U}} \newcommand{\mA}{\mathbf{A}} \newcommand{\mB}{\mathbf{B}} \newcommand{\mC}{\mathbf{C}} \newcommand{\mD}{\mathbf{D}} \newcommand{\mE}{\mathbf{E}} \newcommand{\mF}{\mathbf{F}} \newcommand{\mK}{\mathbf{K}} \newcommand{\mI}{\mathbf{I}} \newcommand{\mM}{\mathbf{M}} \newcommand{\mN}{\mathbf{N}} \newcommand{\mQ}{\mathbf{Q}} \newcommand{\mT}{\mathbf{T}} \newcommand{\mV}{\mathbf{V}} \newcommand{\mW}{\mathbf{W}} \newcommand{\mX}{\mathbf{X}} \newcommand{\ma}{\mathbf{a}} \newcommand{\mb}{\mathbf{b}} \newcommand{\mc}{\mathbf{c}} \newcommand{\md}{\mathbf{d}} \newcommand{\me}{\mathbf{e}} \newcommand{\mn}{\mathbf{n}} \newcommand{\mr}{\mathbf{r}} \newcommand{\mv}{\mathbf{v}} \newcommand{\mw}{\mathbf{w}} \newcommand{\mx}{\mathbf{x}} \newcommand{\mxb}{\mathbf{x_{bet}}} \newcommand{\my}{\mathbf{y}} \newcommand{\mz}{\mathbf{z}} \newcommand{\reel}{\mathbb{R}} \newcommand{\mL}{\bm{\Lambda}} \newcommand{\mnul}{\mathbf{0}} \newcommand{\trap}[1]{\mathrm{trap}(#1)} \newcommand{\Det}{\operatorname{Det}} \newcommand{\adj}{\operatorname{adj}} \newcommand{\Ar}{\operatorname{Areal}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Rum}{\operatorname{Rum}} \newcommand{\diag}{\operatorname{\bf{diag}}} \newcommand{\bidiag}{\operatorname{\bf{bidiag}}} \newcommand{\spanVec}[1]{\mathrm{span}{#1}} \newcommand{\Div}{\operatorname{Div}} \newcommand{\Rot}{\operatorname{\mathbf{Rot}}} \newcommand{\Jac}{\operatorname{Jacobi}} \newcommand{\Tan}{\operatorname{Tan}} \newcommand{\Ort}{\operatorname{Ort}} \newcommand{\Flux}{\operatorname{Flux}} \newcommand{\Cmass}{\operatorname{Cm}} \newcommand{\Imom}{\operatorname{Im}} \newcommand{\Pmom}{\operatorname{Pm}} \newcommand{\IS}{\operatorname{I}} \newcommand{\IIS}{\operatorname{II}} \newcommand{\IIIS}{\operatorname{III}} \newcommand{\Le}{\operatorname{L}} \newcommand{\app}{\operatorname{app}} \newcommand{\M}{\operatorname{M}} \newcommand{\re}{\mathrm{Re}} \newcommand{\im}{\mathrm{Im}} \newcommand{\compl}{\mathbb{C}} \newcommand{\e}{\mathrm{e}} \\$

% What is $\,\displaystyle{\sin\left(3\frac{\pi}{2}\right)}\,$ ?

State the radian numbers that correspond to the angles $30, 60, 120, 135$ and $300$ degrees.

Draw the unit circle in an $(x,y)$ -coordinate system with centre in $(0,0)$ . Draw point on the unit circle corresponding to the arc lengths

$\pi\,,\, \frac{\pi}{3}\,, \,\frac{-\pi}{6}\,, \,-\frac{\pi}{6}\,, \,\frac{7\pi}{12}\,,\,-\frac{3\pi}{2}\,,\,\frac{7\pi}{4}\,.$

To which angular measures in degrees do they correspond?

Use the figure (the blue triangle) for the geometrial determination of the exact values for $\,\displaystyle{\cos\left(\frac{\pi}{4}\right)}\,$ og $\,\displaystyle{\sin\left(\frac{\pi}{4}\right)}\,.$

Determine by the use of symmetry considerations the numbers

$\cos\left(p\,\frac{\pi}{4}\right)\,\,\,\mathrm{and}\,\,\,\sin\left(p\,\frac{\pi}{4}\right) \,\,\,\mathrm{for}\,\,\,p=3, 5, 7, -1, -3, -5, -7\,.$

Given that $\,\displaystyle{\cos\left(\frac{\pi}{6}\right)}=\frac{\sqrt 3}{2}\,$ and $\,\displaystyle{\sin\left(\frac{\pi}{6}\right)}=\frac{1}{2}\,.$ Draw the point

$\, \displaystyle{\left(\cos\left(\frac{\pi}{6}\right)\,,\,\sin\left(\frac{\pi}{6}\right)\right)} \,$

on a unit circle and find by the use of symmetry considerations the numbers

$\cos\left(p\,\frac{\pi}{6}\right)\,\,\,\mathrm{and}\,\,\,\sin\left(p\,\frac{\pi}{6}\right) \,\,\,\mathrm{for}\,\,\,p=2, 4, 5, 7, 8, 10, 11\,.$

For given numbers $\,A,\,b,\,C\,$ and $d$ the graphs for the functions are

$\,x\rightarrow A\cos(b\cdot x)\,\,\,\mathrm{og}\,\,\,x\rightarrow C\sin(d\cdot x)\,$

plotted for $\,x\in \left[\,-\pi\,,\,\pi\,\right]\,$ .

Determine the numbers $\,A,\,b,\,C\,$ and $d\,.$

Given the numbers $z_0=1+i\sqrt{3}\,$ , $z_1=-1+i\sqrt{3}\,$ , $z_2=-1-i\sqrt{3}\,$ and $z_3=1-i\sqrt{3}\,$ .

Find the circle with centre in $0\,$ , where upon the four numbers lie.
Determine $\,\arg(z_0)\,$ , and then state the main argument for $\,z_1\,,\,z_2\,$ and $\,z_3\,$ . What are the polar coordinates for all four numbers?

A student shall find the polar coordinates for the number $\,2-2i\,\,$ . He chooses to use the calculator. By entering

$\sqrt{2^2+(-2)^2}$

he gets the absolute value $\,2\sqrt{2}\,.$ And by entering

$\cos^{-1}\left(\frac 2{2\sqrt{2}}\right)$

he gets the argument $\,\displaystyle{\frac {\pi}4}\,.$

Check the calculations. What is the students error?

Find the absolute value and the main arguments for the following complex numbers:

$-2+2i$ .
$\displaystyle{-\frac{1}{6}+\frac{i}{2\sqrt{3}}}\,$ .

Three complex numbers are given by their polar coordinates like this:

$(4,\,-\pi)\,,\,(2,\,\frac{4\pi}{3})\,,\,(6,\,\frac{21\pi}{4})\,.$

Find the rectangular form of the numbers.

If you do not already have GeoGebra installed on your computer,then do it now. See the link to GeoGebra on the Agenda of today. Open the GeoGebra-sheet komplekseTal. In the sheet four complex numbers $\,z_1,\,z_2,\,z_3\,$ and $\,z_4\,$ are given. Construct the following complex numbers

$\,z_1+z_2\,,\,z_2-z_4\,,\,z_3\cdot z_3\,,\,z_2\cdot z_4\,\,\,\mathrm{og}\,\,\,\frac{z_2}{z_4}\,.$

Draw using the vector tool (in the line box) the position vectors of the found complex numbers. Give them possibly another color. Finally draw the five yellow labels to the corresponding numbers.

Obs: In this exercise you choose your own level of ambition (or go step by step from the easiest to the hardest): Mild: Right click on the sheet and activate Grid. Choose Lock the grid in Configurations $\rightarrow$ Catch the point. Find by use of geometrical considerations the wanted numbers and draw them with the point-tool.\bs Medium: Construct the wanted numbers with special GeoGebra construction tools, e.g. Parallel displace by vector and Reflection in point.\bs Hot: Find the numbers by a pure rule and compasses construction. You find the compasses in the circle toolbox. The ruler construction is done using relevant tools from the line toolbox.

Write the following complex numbers in rectangular form:

$\e^{i\frac{\pi}{2}}$
$3\e^{1+\pi i}$

Given the numbers $z_0=1+i\sqrt{3}\,$ , $z_1=-\sqrt{3}+i\,$ , $z_2=-1-i\sqrt{3}\,\,$ and $\,z_3=\sqrt{3}-i\,$ .

State the four numbers in exponential form.
Show that a binomial equation

$\,z^4=w\,$

exists to which all four numbers are a solution.

Given $w=1-i\,$ .

Determine $|\,w\,|$ and $\arg(w)\,$ .
Determine $|\,\e^w\,|$ and $\arg(\e^w)\,$ .

Given the numbers $\,w_1=1\,,\,w_2=\e\,,\,w_3=i\,$ and $\,w_4=2i\,$ . Determine all solutions to the equations

$\e^z=w_n$

where $\,n=1\,.\,.\,4\,$ .

Determine all solutions to the equation

$(\e^z-1)(\e^z-i)=0\,.$

Show that $\,\e^z\neq0\,$ for all $\,z\in\mathbb C\,$ .

Here are the so-called formulas for double angles:

$\sin(2v)=2\sin(v)\cos(v)\,\,\,\mathrm{and} \,\,\,\cos(2v)=(\cos(v))^2-(\sin(v))^2\,.$

Use the latter (together with the idiot rule) to determine $\cos\left(\frac{\pi}{8}\right)\,$ and $\sin\left(\frac{\pi}{12}\right)\,.$

Hint: Start from the known exact values for cosine of $\,\frac{\pi}4\,$ and sine of $\,\frac{\pi}6\,$ , respectively.

Use the first result in a) to find $\sin\left(\frac{\pi}{8}\right)\,,\cos\left(\frac{3\pi}{8}\right)\,$ and $\,\sin\left(\frac{3}{8}\pi\right)\,$ .

Similarly use the second result in a) to find cosine and sine of interesting angles of the form $\,\frac{p\pi}{12}\,.$