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<title>Matematik<!--#include virtual="/shtml/titleSuffix.shtml"--></title>
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<p class="heading">Matematik</p>
<div class="section" id="rules">
<p class="heading">Regneregler</p>
<div class="group">
<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>
</div>
<div class="group">
<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>
</div>
<div class="group">
<p class="math">\(\frac xy=z\)</p>
<p class="math">\(zy=x\)</p>
<p class="math">\(\frac {x}{z}=y\)</p>
</div>
<div class="group">
<p class="math">\(x^y=z\)</p>
<p class="math">\(\sqrt[y] {z}=x\)</p>
<p class="math">\(log_x(z)=y\)</p>
</div>
<div class="group">
<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">\(x^{\frac nm}=\sqrt[m] {x^n}\)</p>
</div>
<div class="group">
<p class="math">\(\frac xy+n=\frac {x+n y}{y}\)</p>
<p class="math">\(\frac xy+\frac ab=\frac {x b+ay}{yb}\)</p>
<p class="math">\(\frac xyn=\frac {xn}{y}\)</p>
<p class="math">\(\frac xy\frac ab=\frac {x a}{y b}\)</p>
<p class="math">\(\frac {x}{\frac ab}=\frac {xb}{a}\)</p>
<p class="math">\(\frac {\frac xy}z=\frac {x}{yz}\)</p>
<p class="math">\(\frac {\frac xy}{\frac ab}=\frac {xb}{ya}\)</p>
</div>
<div class="group">
<p class="math">\((x^a)^b=x^{ab}\)</p>
<p class="math">\(x^ay^a=(xy)^a\)</p>
<p class="math">\(\frac {x^a}{y^a}=(\frac xy)^a\)</p>
<p class="math">\(x^ax^b=x^{a+b}\)</p>
<p class="math">\(\frac {x^a}{x^b}=x^{a-b}\)</p>
</div>
</div>
<div class="section" id="equations">
<p class="heading">Ligninger</p>
<p>Andengrads:</p>
<div class="group">
<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>
<div class="section" id="functions">
<p class="heading">Funktioner</p>
<div class="group">
<p class="math">\(y=f(x)\)</p>
<p class="math">\(x=f^{-1}(y)\)</p>
</div>
<div class="group">
<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>
</div>
<div class="group">
<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>
</div>
<div class="group">
<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>
</div>
<div class="group">
<p>Andengrads:</p>
<p class="math">\(f(x)=ax^2+bx+c\)</p>
</div>
</div>
<div class="section" id="trigonometry">
<p class="heading">Trigonometri</p>
<div class="group">
<p class="math">\(modliggende_{\alpha}=hosliggende_{\beta}=a\)</p>
<p class="math">\(hosliggende_{\alpha}=modliggende_{\beta}=b\)</p>
<p class="math">\(hypotenuse=modliggende_{\gamma}=c\)</p>
</div>
<div class="group">
<p>I en retvinklet trekant:</p>
<p class="math">\(sin(\theta)=\frac {modliggende_{\theta}}{hypotenuse_{\theta}}\)</p>
<p class="math">\(cos(\theta)=\frac {hosliggende_{\theta}}{hypotenuse_{\theta}}\)</p>
<p class="math">\(tan(\theta)=\frac {modliggende_{\theta}}{hosliggende_{\theta}}\)</p>
<p class="math">\(csc(\theta)=\frac 1{sin(\theta)}=\frac {hypotenuse_{\theta}}{modliggende_{\theta}}\)</p>
<p class="math">\(sec(\theta)=\frac 1{cos(\theta)}=\frac {hypotenuse_{\theta}}{hosliggende_{\theta}}\)</p>
<p class="math">\(cot(\theta)=\frac 1{tan(\theta)}=\frac {hosliggende_{\theta}}{modliggende_{\theta}}\)</p>
</div>
<div class="group">
<p>I en retvinklet trekant:</p>
<p class="math">\(sin^{-1}(\frac {modliggende_{\theta}}{hypotenuse_{\theta}})=\theta\)</p>
<p class="math">\(cos^{-1}(\frac {hosliggende_{\theta}}{hypotenuse_{\theta}})=\theta\)</p>
<p class="math">\(tan^{-1}(\frac {modliggende_{\theta}}{hosliggende_{\theta}})=\theta\)</p>
<p class="math">\(csc^{-1}(\frac {hypotenuse_{\theta}}{modliggende_{\theta}})=\theta\)</p>
<p class="math">\(sec^{-1}(\frac {hypotenuse_{\theta}}{hosliggende_{\theta}})=\theta\)</p>
<p class="math">\(cot^{-1}(\frac {hosliggende_{\theta}}{modliggende_{\theta}})=\theta\)</p>
</div>
<div class="group">
<p>Forkortelser:</p>
<p class="math">\(sin=sinus\)</p>
<p class="math">\(cos=cosinus\)</p>
<p class="math">\(tan=tangens\)</p>
<p class="math">\(csc=cosekant\)</p>
<p class="math">\(sec=sekant\)</p>
<p class="math">\(cot=cotangens\)</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">\(arccsc=csc^{-1}\)</p>
<p class="math">\(arcsec=sec^{-1}\)</p>
<p class="math">\(arccot=cot^{-1}\)</p>
</div>
<div class="group">
<p class="math">\(deg(rad)=\frac {rad \cdot 180}{\pi}\)</p>
<p class="math">\(rad(deg)=\frac {deg \cdot \pi}{180}\)</p>
</div>
<div class="group">
<p class="math">\(\Theta(n)=\pi(n-2)\)</p>
<p class="note">hvori <span class="math">\({\Theta}(n)\)</span> er vinkelsummen af <span class="emphasis">n</span>-gonen.</p>
<p class="math">\(\Theta(3)=\pi\)</p>
</div>
<div class="group">
<p class="math">\(\alpha=\pi-\beta-\gamma\)</p>
<p class="math">\(\beta=\pi-\alpha-\gamma\)</p>
<p class="math">\(\gamma=\pi-\alpha-\beta\)</p>
<p>I en retvinklet trekant:</p>
<p class="math">\(\alpha=sin^{-1}(\frac ac)=cos^{-1}(\frac bc)=tan^{-1}(\frac ab)\)</p>
<p class="math">\(\beta=sin^{-1}(\frac bc)=cos^{-1}(\frac ac)=tan^{-1}(\frac ba)\)</p>
<p class="math">\(\gamma=\frac {\pi}2\)</p>
<p>I en regulær trekant:</p>
<p class="math">\(\alpha=\beta=\gamma=\frac {\pi}3\)</p>
</div>
<div class="group">
<p>I en retvinklet trekant:</p>
<p class="math">\(a=c \cdot sin(\alpha)=c \cdot cos(\beta)=b \cdot tan(\alpha)=b \cdot cot(\beta)=\sqrt[2] {c^2-b^2}\)</p>
<p class="math">\(b=c \cdot sin(\beta)=c \cdot cos(\alpha)=a \cdot tan(\beta)=a \cdot cot(\alpha)=\sqrt[2] {c^2-a^2}\)</p>
<p class="math">\(c=a \cdot csc(\alpha)=b \cdot csc(\beta)=a \cdot sec(\beta)=b \cdot sec(\alpha)=\sqrt[2] {a^2+b^2}\)</p>
<p>I en retvinklet trekant, hvori kateterne har samme længde:</p>
<p class="math">\(a=b=\sqrt[2] {\frac {c^2}2}\)</p>
<p>I en regulær trekant:</p>
<p class="math">\(a=b=c\)</p>
</div>
<div class="group">
<p class="math">\(O=a+b+c\)</p>
<p class="math">\(A=\frac {bh}2\)</p>
<p class="note">hvori <span class="emphasis">b</span> er bredten af trekanten og <span class="emphasis">h</span> er dens højde.</p>
<p>Mellem to ligedannede trekanter:</p>
<p class="math">\(\alpha_1=\alpha_0\)</p>
<p class="math">\(\beta_1=\beta_0\)</p>
<p class="math">\(\gamma_1=\gamma_0\)</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>
<div class="section" id="constants">
<p class="heading">Konstanter</p>
<table>
<thead>
<tr>
<th>Navn</th>
<th>Symbol</th>
</tr>
</thead>
<tbody>
<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 (<span class="emphasis">pi</span>)</td>
<td class="math">\(\pi\)</td>
</tr>
<tr>
<td><span class="emphasis">tau</span></td>
<td class="math">\(\tau\)</td>
</tr>
<tr>
<td><sub>den </sub>gyldne ratio</td>
<td class="math">\(\varphi\)</td>
</tr>
</tbody>
</table>
<div class="group">
<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">\(\varphi=\frac {1+\sqrt[2] 5}2 \approx \frac {1\ 618\ 033\ 989}{10^9}\)</p>
</div>
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