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<!DOCTYPE html>
<html lang="da">
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<head>
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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>
<br />
<p class="math">\(xy = z\)</p>
<p class="math">\(\frac{z}{y} = x\)</p>
<p class="math">\(\frac{z}{x} = y\)</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">\(x ^ n = \frac{1}{x ^ {-n}}, x \lt 0\)</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">\(y = ax ^ 2 + bx + c\)</p>
<p class="math">\(x = \frac{-b \pm \sqrt[2]{d}}{2a}\)</p>
<p class="math">\(d = b ^ 2 - 4ac\)</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>Kvadratisk (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">\(deg(rad) = \frac{rad\pi}{180}\)</p>
<p class="math">\(rad(deg) = \frac{deg180}{\pi}\)</p>
<br />
<p class="math">\(vinkelsum(x) = (x - 2)\pi\)</p>
<p class="math">\(vinkelsum(3) = (3 - 2)\pi = \pi\)</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">\(\angle C = vinkelsum(3) - \angle A - \angle B\)</p>
<p>I en regulær trekant:</p>
<p class="math">\(\angle A = \angle B = \angle C\)</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><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">\(\phi = \frac{1 + \sqrt[2]{5}}{2} \approx \frac{1\ 618\ 033\ 989}{10 ^ 9}\)</p>
</div>
</div>
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</html>
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