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Exercise 1: Today’s Wetware Exercise

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

Exercise 2: Angular Measures in Degrees and Radians

A

State the radian values that correspond to the angles $30^\circ, 60^\circ, 120^\circ, 135^\circ$ and $300^\circ$.

B

Draw the unit circle in an $(x,y)$ coordinate system centered at $(0,0)$. Draw points on the unit circle corresponding to the arc lengths

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

Which degree values do these angles in radians correspond to?

Exercise 3: Sine and Cosine

A

Use the figure (the blue triangle) to geometrically derive the exact values of $\,\displaystyle{\cos\left(\frac{\pi}{4}\right)}\,$ and $\,\displaystyle{\sin\left(\frac{\pi}{4}\right)}\,.$

B

Compute via symmetry considerations the values

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

C

Given $\,\displaystyle{\cos\left(\frac{\pi}{6}\right)}=\frac{\sqrt 3}{2}\,$ 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

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

D

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

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

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

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

Exercise 4: Polar coordinates

A

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

  1. State the circle centered in $0$ at which the four numbers are located.

  2. Compute $\,\arg(z_0)\,$ and then state the principal argument of $\,z_1\,,\,z_2\,$ and $\,z_3\,$. What are the polar coordinates of all four numbers?

B

A student needs to find the polar coordinates for the number $\,2-2i\,\,$. He chooses to use a 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 error has the student made?

C

Find the modulus and the principal argument of the following complex numbers:

  1. $-2+2i$

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

D

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

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

Determine the rectangular form of the numbers.

Exercise 5: The Exponential Form of Complex Numbers

A

Write the following complex numbers on rectangular form:

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

B

The numbers $z_0=1+i\sqrt{3}\,$, $z_1=-\sqrt{3}+i\,$, $z_2=-1-i\sqrt{3}\,\,$ and $\,z_3=\sqrt{3}-i\,$ are given.

  1. Convert the four numbers to exponential form.

  2. Show that a binomial equation

$$\,z^4=w\,,$$

were $w$ is a complex scalar, exists to which all four numbers are a solution.

Exercise 6: The Complex Exponential Function

A

The number $\,w=1-i\,$ is given.

  1. Compute $|\,w\,|$ and $\hbox{Arg}(w)\,$.

  2. Compute $|\,\e^w\,|$ and $\hbox{Arg}(\e^w)\,$.

B

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 $\,n=1\ldots4\,$.

Determine all solutions to the equation

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

C

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

Exercise 7: Geometric Addition and Multiplication

A

It is now time to install GeoGebra on your computer or to access it via a browser. See the link to GeoGebra on the Agenda of today.

When the software is ready, open the GeoGebra sheet komplekseTal. In this sheet, four complex numbers $\,z_1\,,\,z_2\,,\,z_3\,$ and $\,z_4\,$ are given. Construct the following complex numbers:

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

Draw using the vector tool (in the line box) the position vectors of the found complex numbers. You might want to colour them differently. Finally, draw the five yellow labels to the corresponding numbers.

Note: In this exercise you may 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 via geometric considerations the wanted numbers and draw them with the point tool.
  • Medium: Construct the wanted numbers with dedicated GeoGebra construction tools, e.g. Parallel displace by vector and Reflection in point.
  • Challenging: Find the numbers purely from ruler and compass construction. You’ll find the compass tools in the circle toolbox. A ruler construction is done using relevant tools from the line toolbox.

Exercise 8: Double Angles (Advanced)

A

The formulas of double angles as they are called are the following:

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

Use the latter (along with the dummy rule $\cos(v)^2+\sin(v)^2 =1$) to determine $\cos\left(\frac{\pi}{8}\right)\,$ and $\sin\left(\frac{\pi}{12}\right)\,.$

B

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

C

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