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%tandborste2.png Reduce $\,\displaystyle{\frac{2^{-1}\cdot 2^4\cdot (2^3)^{-2}}{2^{-5}}\,\,.}\,$

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%$4$
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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 is to an acute angle is the subtending 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 often called the idiot rule. Which mathematical explanations can be given for this unflattering name?

A

State the numbers$\,\displaystyle{\mathrm{Arccos}\left(\frac{1}{2}\right),\,\mathrm{Arcsin}\left(-\frac{\sqrt 3}{2}\right)\,\,\,\mathrm{and}\,\,\, \mathrm{Arcsin}(1)}\,.$

Given the sets $\,A=\left{x\in\reel\,|\,x\in \left[\,0\,,\,2\pi\,\right]\right}\,$ and $\,B=\left{x\in\reel\,|\,x\in \left[\,-\pi\,,\,\pi\,\right]\right}\,.$

B

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

C

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

D

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

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%Løs ligningen $\,\displaystyle{\cos(x)=-\frac 1 5}\,$ inden for hver af mængderne $\,A,\,B\,$ and $\,\Bbb R\,.$ 
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A

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

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

A

Given $z=1+i\,.$ Determine the absolute value and the main argument for $z$, and using this determine the absolute value and main argument for $z^2\,,$ $z^5\,,$ $z^8\,,$ and $z^{-10}\,$. Finally state the rectangular form for $z^2\,,$ $z^5\,,$ $z^8\,,$ and $z^{-10}\,$.

A

Solve the binomial equations

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

Sketch the solutions in the complex number plane.

B

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.

A

A set has at $t=0 $ the magnitude $b$ and grows with $\,20\%\,$ 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.

B

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

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

C

The Figures shows the graph for three exponential functions. State their base, and write each of them in 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

%### Omvendte funktioner (advanced)\label{U2LD6}
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%Den naturlige logaritme indføres som den omvendte funktion til funktionen 

$$\,x\rightarrow \e^x\,,\,\,x\in \reel\,.$$

 Bevis regnereglerne
%

$$\ln(a\cdot b)=\ln(a)+\ln(b)\,\,\,\mathrm{og}\,\,\,\ln(c^n)=n\ln(c)$$


%hvor $a,\,b$ og $c$ er positive reelle tal og $n$ er et helt tal.
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