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diff --git a/html/encyclopedia/mathematics.html b/html/encyclopedia/mathematics.html index 948183e..888361f 100644 --- a/html/encyclopedia/mathematics.html +++ b/html/encyclopedia/mathematics.html @@ -11,147 +11,147 @@ <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">\(\frac{x}{y} = z\)</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">\(x^n = \frac{1}{x^{-n}}, x \lt 0\)</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> + <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">\(\frac{x}{y}=z\)</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">\(x^n=\frac{1}{x^{-n}}, x \lt 0\)</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">\(ax^2 + bx + c = 0\)</p> - <p class="math">\(x = \frac{-b \pm \sqrt[2]{d}}{2a}\)</p> - <p class="math">\(d = b^2 - 4ac\)</p> + <p class="math">\(ax^2+bx+c=0\)</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> + <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> + <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> + <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> + <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> + <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> + <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 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">\(vinkelsum(x)=(x-2)\pi\)</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 class="math">\(\angle A=\angle B=\angle C\)</p> <p>I en retvinklet trekant:</p> - <p class="math">\(\angle C = \frac{\pi}{2}\)</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 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 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 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> + <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 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> + <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> @@ -187,10 +187,10 @@ </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">\(e=\sum_{n=0}^\infty \frac{1}{n!} \approx \frac{2\ 718\ 281\ 828}{10^9}\)</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> + <p class="math">\(\phi=\frac{1+\sqrt[2]{5}}{2} \approx \frac{1\ 618\ 033\ 989}{10^9}\)</p> </div> </div> <!--#include virtual="/include/pgftr.shtml"--> |