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Exercise 1: Today’s Wetware Exercise

Reduce $\,\displaystyle{\frac{2^{-1}\cdot 2^4\cdot (2^3)^{-2}}{2^{-5}}\,\,.}\,$

Exercise 2: Right-Angled Triangles

A

Given an arbitrary right-angled triangle. Show from the definition of cosine and sine that:

  • The cosine of an acute angle is the adjacent side divided by the hypotenuse.

  • The sine of an acute angle is the opposing side divided by the hypotenuse.

B

A right-angled triangle has the sides$\,a=1\,$, $b=\sqrt{3}\,$ and $c=2\,.$ What are the exact angles expressed in radians?

C

The important trigonometric formula

$$\,\cos(v)^2+\sin(v)^2=1\,$$

is in Denmark typically called idiotreglen, translating directly to the idiot rule or the dummy rule. Which mathematical explanations can be given for this unflattering name?

Exercise 3: Arccos, Arcsine and Trigonometric Equations

D

Compute the numbers

$$\,\displaystyle{\mathrm{arccos}\left(\frac{1}{2}\right),\,\mathrm{arcsin}\left(-\frac{\sqrt 3}{2}\right)\quad\mathrm{and}\quad \mathrm{arcsin}(1)}\,.$$

Let $A=\{x\in{\mathbb R}|\,x\in \left[\,0\,,\,2\pi\,\right]\}$ and $B=\{x\in\reel\,|\,x\in \left[\,-\pi\,,\,\pi\,\right]\}$.

E

Solve the equation $\,\displaystyle{\cos(x)=\frac{1}{2}}\,$ within each of the sets $\,A,\,B\,$ and $\,\Bbb R\,.$

F

Solve the equation $\,\displaystyle{\sin(x)=-\frac{\sqrt 3}{2}}\,$ within each of the sets $\,A,\,B\,$ and $\,\Bbb R\,.$

G

Solve the equation $\,\displaystyle{\mathrm e^{\,i\cdot v}= \frac{1}{2}-\frac{\sqrt 3}{2}\,i\,}\,$ within each of the sets $\,A\,$ and $\,B\,.$

Exercise 4: Computations with Powers

H

Rewrite the following expressions to numbers of the form $\,a^p\,$ where $\,a\,$ is a positve real number and $\,p\in \mathbb Z$:

$$ 3^2\cdot 3^{3}\,,\,\,(5^8)^{-2}\,,\,\,3^2\cdot 3^{-5}\,,\,\,\frac{4^{1.3}}{4^{2.3}} \,,\,\,\left(\frac{1}{2}\right)^5\cdot 6^5 \,,\,\,\frac{5^3}{0.5^3}\,. $$

Exercise 5: Powers of Complex Numbers

I

Given $z=1+i\,,$ compute the modulus and principal argument of $z$. Use this result to determine the modulii and principal arguments of $z^2\,,$ $z^5\,,$ $z^8\,,$ and $z^{-10}\,$. Finally, provide the rectangular form of $z^2\,,$ $z^5\,,$ $z^8\,,$ and $z^{-10}\,$.

Exercise 6: Binomial Equations

J

Solve the binomial equations

$${z}^2 = -4\,\,,\,\,{z}^2 = i\,\,\,\mathrm{and}\,\,\,{z}^2 = 1-i\,.$$

Sketch the solutions in the complex number plane.

K

Solve the binomial equations

  1. $z^3=1$

  2. $z^3=i$

  3. $z^3=1+i$

and sketch the solutions in the complex number plane.

Exercise 7: Exponential Growth (Highschool Recap)

L

A set has, at $t=0$, the magnitude $b$ and grows $\,20\,\%\,$ per time unit. Determine the number $a$ such that the functional expression

$$\,f(t)=b\cdot a^t\,,\,\,t\in\reel$$

can be considered to be a model for the growth.

M

Determine the growth rate and the percentage growth per time unit for the exponentially growing/decreasing functions given by the functional expressions:

$$f(t)=2^t\,\,\,\mathrm{and}\,\,\,h(t)=0.5^{\,2-t}\,,\,\,t\in \reel.$$

N

The figure below shows the graph of three exponential functions. State their base, and write each of them on the form

$$\,x\rightarrow \e^{kx}\,,\,\,x\in \reel$$

where $\e$ is the base for the natural exponential function and $k$ is a real number.

eksponentialfunktioner.png