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<!DOCTYPE html>
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<body>
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<div class="content">
<p class="pageTitle">Leksikon<sub> Matematik</sub></p>
<div class="section" id="rules">
<p class="large">Regneregler</p>
<p class="math">\(x+y=z\)</p>
<p class="math">\(z-y=x\)</p>
<p class="math">\(z-x=y\)</p>
<p class="math">\(\sum_{i=a}^b i=a+(a \pm 1) + \cdots +(b \pm 1)+b\)</p>
<br />
<p class="math">\(xy=z\)</p>
<p class="math">\(\frac{z}{y}=x\)</p>
<p class="math">\(\frac{z}{x}=y\)</p>
<p class="math">\(\prod_{i=a}^b i=a(a \pm 1) \cdots (b \pm 1)b\)</p>
<p class="math">\(n!=\prod_{i=1}^n i,n \gt 0\)</p>
<p class="math">\(0!=1\)</p>
<br />
<p class="math">\(\frac{x}{y}=z\)</p>
<p class="math">\(zy=x\)</p>
<p class="math">\(\frac{x}{z}=y\)</p>
<br />
<p class="math">\(x^y=z\)</p>
<p class="math">\(\sqrt[y]{z}=x\)</p>
<p class="math">\(log_{x}(z)=y\)</p>
<br />
<p class="math">\(x^n=\prod_{i=1}^n x, n \gt 0\)</p>
<p class="math">\(x^n=\frac{1}{x^{-n}}, n \lt 0\)</p>
<p class="math">\(x^0=1\)</p>
<p class="math">\(x^{\frac{a}{b}}=\sqrt[b]{x^a}\)</p>
<br />
<p class="math">\(\frac{x}{y}=x\frac{1}{y}\)</p>
<p class="math">\(\frac{x}{y}+n=\frac{x+n y}{y}\)</p>
<p class="math">\(\frac{x}{y}+\frac{a}{b}=\frac{x b+ay}{yb}\)</p>
<p class="math">\(\frac{x}{y}n=\frac{xn}{y}\)</p>
<p class="math">\(\frac{x}{y}\frac{a}{b}=\frac{x a}{y b}\)</p>
<p class="math">\(\frac{x}{\frac{a}{b}}=\frac{xb}{a}\)</p>
<p class="math">\(\frac{\frac{x}{y}}{z}=\frac{x}{yz}\)</p>
<p class="math">\(\frac{\frac{x}{y}}{\frac{a}{b}}=\frac{xb}{ya}\)</p>
<br />
<p class="math">\(x^ax^b=x^{a+b}\)</p>
<p class="math">\(\frac{x^a}{x^b}=x^{a-b}\)</p>
<p class="math">\(x^ay^a=(xy)^a\)</p>
<p class="math">\(\frac{x^a}{y^a}=(\frac{x}{y})^a\)</p>
<p class="math">\((x^a)^b=x^{ab}\)</p>
</div>
<div class="section" id="equations">
<p class="large">Ligninger</p>
<p>Andengrads:</p>
<p class="math">\(ax^2+bx+c=0\)</p>
<p class="math">\(d=b^2-4ac\)</p>
<p class="math">\(x=\frac{-b \pm \sqrt[2]{d}}{2a}\)</p>
</div>
<div class="section" id="functions">
<p class="large">Funktioner</p>
<p class="math">\(y=f(x)\)</p>
<p class="math">\(x=f^{-1}(y)\)</p>
<br />
<p>Lineær:</p>
<p class="math">\(f(x)=ax+b\)</p>
<p class="math">\(a=\frac{y_1-y_0}{x_1-x_0}\)</p>
<p class="math">\(b=y-ax\)</p>
<p class="math">\(f(0)=b\)</p>
<br />
<p>Eksponentiel:</p>
<p class="math">\(f(x)=ba^x\)</p>
<p class="math">\(a=\sqrt[x_1-x_0]{\frac{y_1}{y_{0}}}\)</p>
<p class="math">\(b=\frac{y}{a^x}\)</p>
<p class="math">\(f(0)=b\)</p>
<br />
<p>Potens:</p>
<p class="math">\(f(x)=bx^a\)</p>
<p class="math">\(a=\frac{log_n(y_1)-log_n(y_0)}{log_n(x_1)-log_n(x_1)}\)</p>
<p class="math">\(b=\frac{y}{x^a}\)</p>
<p class="math">\(f(0)=0\)</p>
<p class="math">\(f(1)=b\)</p>
<br />
<p>Andengrads:</p>
<p class="math">\(f(x)=ax^2+bx+c\)</p>
</div>
<div class="section" id="trigonometry">
<p class="large">Trigonometri</p>
<p class="math">\(modliggende_{\angle A}=hosliggende_{\angle B}=a\)</p>
<p class="math">\(hosliggende_{\angle A}=modliggende_{\angle B}=b\)</p>
<p class="math">\(hypotenuse=modliggende_{\angle C}=c\)</p>
<br />
<p class="math">\(sin(\theta)=\frac{modliggende}{hypotenuse}\)</p>
<p class="math">\(cos(\theta)=\frac{hosliggende}{hypotenuse}\)</p>
<p class="math">\(tan(\theta)=\frac{modliggende}{hosliggende}\)</p>
<p class="math">\(cot(\theta)=\frac{hosliggende}{modliggende}\)</p>
<p class="math">\(csc(\theta)=\frac{hypotenuse}{modliggende}\)</p>
<p class="math">\(sec(\theta)=\frac{hypotenuse}{hosliggende}\)</p>
<br />
<p class="math">\(sin^{-1}(\frac{modliggende}{hypotenuse})=\theta\)</p>
<p class="math">\(cos^{-1}(\frac{hosliggende}{hypotenuse})=\theta\)</p>
<p class="math">\(tan^{-1}(\frac{modliggende}{hosliggende})=\theta\)</p>
<p class="math">\(cot^{-1}(\frac{hosliggende}{modliggende})=\theta\)</p>
<p class="math">\(csc^{-1}(\frac{hypotenuse}{modliggende})=\theta\)</p>
<p class="math">\(sec^{-1}(\frac{hypotenuse}{hosliggende})=\theta\)</p>
<br />
<p>Forkortelser:</p>
<p class="math">\(sin=sinus\)</p>
<p class="math">\(cos=cosinus\)</p>
<p class="math">\(tan=tangens\)</p>
<p class="math">\(cot=cotangens\)</p>
<p class="math">\(csc=cosekant\)</p>
<p class="math">\(sec=sekant\)</p>
<p class="math">\(arcsin=sin^{-1}\)</p>
<p class="math">\(arccos=cos^{-1}\)</p>
<p class="math">\(arctan=tan^{-1}\)</p>
<p class="math">\(arccot=cot^{-1}\)</p>
<p class="math">\(arcsec=sec^{-1}\)</p>
<p class="math">\(arccsc=csc^{-1}\)</p>
<br />
<p class="math">\(deg(rad)=\frac{rad \cdot 180}{\pi}\)</p>
<p class="math">\(rad(deg)=\frac{deg \cdot \pi}{180}\)</p>
<br />
<p class="math">\(\sum \theta=(n-2)\pi\)</p>
<p>... hvori <i>n</i> er antallet af vinkler.</p>
<p class="math">\(\sum \theta=(3-2)\pi=\pi\)</p>
<br />
<p class="math">\(\angle A=sin^{-1}(\frac{a}{c})=cos^{-1}(\frac{b}{c})=tan^{-1}(\frac{a}{b})=vinkelsum(3)-\angle B-\angle C\)</p>
<p class="math">\(\angle B=sin^{-1}(\frac{b}{c})=cos^{-1}(\frac{a}{c})=tan^{-1}(\frac{b}{a})=vinkelsum(3)-\angle A-\angle C\)</p>
<p class="math">\(\angle C=(\sum \theta)-\angle A-\angle B\)</p>
<p>I en regulær trekant:</p>
<p class="math">\(\angle A=\angle B=\angle C=\frac{\pi}{3}\)</p>
<p>I en retvinklet trekant:</p>
<p class="math">\(\angle C=\frac{\pi}{2}\)</p>
<br />
<p class="math">\(a=c \cdot sin(\angle A)=c \cdot cos(\angle B)=b \cdot tan(\angle A)=b \cdot cot(\angle B)\)</p>
<p class="math">\(b=c \cdot sin(\angle B)=c \cdot cos(\angle A)=a \cdot tan(\angle B)=a \cdot cot(\angle A)\)</p>
<p class="math">\(c=a \cdot csc(\angle A)=b \cdot csc(\angle B)=a \cdot sec(\angle B)=b \cdot sec(\angle A)\)</p>
<p>I en regulær trekant:</p>
<p class="math">\(a=b=c\)</p>
<p>I en retvinklet trekant:</p>
<p class="math">\(a=\sqrt[2]{c-b^2}\)</p>
<p class="math">\(b=\sqrt[2]{c-a^2}\)</p>
<p class="math">\(c=\sqrt[2]{a^2+b^2}\)</p>
<p>I en retvinklet trekant, hvori kateterne har samme længde:</p>
<p class="math">\(a=b=\sqrt[2]{\frac{c^2}{2}}\)</p>
<br />
<p class="math">\(O=a+b+c\)</p>
<p class="math">\(A=\frac{b h}{2}\)</p>
<p>Mellem to ligedannede trekanter:</p>
<p class="math">\(\angle A_1=\angle A_0\)</p>
<p class="math">\(\angle B_1=\angle B_0\)</p>
<p class="math">\(\angle C_1=\angle C_0\)</p>
<p class="math">\(k=\frac{a_1}{a_0}=\frac{b_1}{b_0}=\frac{c_1}{c_0}\)</p>
<p class="math">\(a_1=a_0 k\)</p>
<p class="math">\(b_1=b_0 k\)</p>
<p class="math">\(c_1=c_0 k\)</p>
<p class="math">\(O_1=O_0 k\)</p>
<p class="math">\(A_1=A_0 k^2\)</p>
</div>
<div class="section" id="constants">
<p class="large">Konstanter</p>
<table>
<tr>
<th>Navn</th>
<th>Symbol</th>
</tr>
<tr>
<td>Pythagoras' konstant</td>
<td class="math">\(\sqrt[2]{2}\)</td>
</tr>
<tr>
<td>Theodorus' konstant</td>
<td class="math">\(\sqrt[2]{3}\)</td>
</tr>
<tr>
<td>Eulers tal</td>
<td class="math">\(e\)</td>
</tr>
<tr>
<td><sub>den </sub>imaginære enhed</td>
<td class="math">\(i\)</td>
</tr>
<tr>
<td>Arkimedes' konstant (<i>pi</i>)</td>
<td class="math">\(\pi\)</td>
</tr>
<tr>
<td><i>tau</i></td>
<td class="math">\(\tau\)</td>
</tr>
<tr>
<td><sub>den </sub>gyldne ratio</td>
<td class="math">\(\phi\)</td>
</tr>
</table>
<p class="math">\(\sqrt[2]{2} \approx \frac{1\ 414\ 213\ 562}{10^9}\)</p>
<p class="math">\(\sqrt[2]{3} \approx \frac{1\ 732\ 050\ 808}{10^9}\)</p>
<p class="math">\(e=\sum_{n=0}^\infty \frac{1}{n!} \approx \frac{2\ 718\ 281\ 828}{10^9}\)</p>
<p class="math">\(i=\sqrt[2]{-1}\)</p>
<p class="math">\(\pi \approx \frac{3\ 141\ 592\ 654}{10^9}\)</p>
<p class="math">\(\tau=2\pi \approx \frac{6\ 283\ 185\ 307}{10^9}\)</p>
<p class="math">\(\phi=\frac{1+\sqrt[2]{5}}{2} \approx \frac{1\ 618\ 033\ 989}{10^9}\)</p>
</div>
</div>
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