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diff --git a/html/encyclopedia/mathematics.html b/html/encyclopedia/mathematics.html index 2cf20d4..426e717 100644 --- a/html/encyclopedia/mathematics.html +++ b/html/encyclopedia/mathematics.html @@ -19,27 +19,61 @@ <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">\(x ^ y = z\)</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> + <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">\(x ^ {a} x ^ {b} = x ^ {a + b}\)</p> - <p class="math">\(\frac{x ^ {a}}{x ^ {b}} = x ^ {a - b}\)</p> - <p class="math">\(x ^ {a} y ^ {a} = (x y) ^ {a}\)</p> - <p class="math">\(\frac{x ^ {a}}{y ^ {a}} = (\frac{x}{y}) ^ {a}\)</p> - <p class="math">\((x ^ {a}) ^ {b} = x ^ {a b}\)</p> - <br /> - <p class="math">\(\frac{x}{y} = x \frac{1}{y}\)</p> + <p class="math">\(\frac{x}{y} = x \cdot \frac{1}{y}\)</p> <p class="math">\(\frac{x}{y} + n = \frac{x + n y}{y}\)</p> <p class="math">\(\frac{x}{y} + \frac{a}{b} = \frac{x b + a y}{y b}\)</p> <p class="math">\(\frac{x}{y} n = \frac{x n}{y}\)</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">\(\frac{\frac{x}{y}}{\frac{a}{b}} = \frac{x b}{y a}\)</p> + <br /> + <p class="math">\(x ^ a x ^ b = x ^ {a + b}\)</p> + <p class="math">\(\frac{x ^ a}{x ^ b} = x ^ {a - b}\)</p> + <p class="math">\(x ^ a y ^ a = (x y) ^ a\)</p> + <p class="math">\(\frac{x ^ a}{y ^ a} = (\frac{x}{y}) ^ a\)</p> + <p class="math">\((x ^ a) ^ b = x ^ {a b}\)</p> + </div> + <div class="section" id="equations"> + <p class="large">Ligninger</p> + <p>Andengrads:</p> + <p class="math">\(y = a x ^ 2 + b x + c\)</p> + <p class="math">\(x = \frac{-b \pm \sqrt[2]{d}}{2a}\)</p> + <p class="math">\(d = b ^ 2 - 4 a c\)</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">\(f(x) = a x + b\)</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">\(b = f(0) = 0a + b\)</p> + <br /> + <p>Eksponentiel:</p> + <p class="math">\(f(x) = b a ^ x\)</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">\(b = f(0) = b a ^ 0 = 1b\)</p> + <br /> + <p>Potens:</p> + <p class="math">\(f(x) = b x ^ a\)</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">\(b = f(1) = b \cdot 1 ^ a = 1b\)</p> + <br /> + <p>Kvadratisk (andengrads):</p> + <p class="math">\(f(x) = a x ^ 2 + b x + c\)</p> </div> <div class="section" id="trigonometry"> <p class="large">Trigonometri</p> @@ -47,14 +81,6 @@ <p class="math">\(hosliggende_{\angle A} = modliggende_{\angle B} = b\)</p> <p class="math">\(hypotenuse = modliggende_{\angle C} = c\)</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> - <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> @@ -62,7 +88,6 @@ <p class="math">\(csc(\theta) = \frac{hypotenuse}{modliggende}\)</p> <p class="math">\(sec(\theta) = \frac{hypotenuse}{hosliggende}\)</p> <br /> - <p class="math">\(x = f ^ {-1} (f(x))\)</p> <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> @@ -70,6 +95,13 @@ <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> @@ -77,11 +109,11 @@ <p class="math">\(arcsec = sec ^ {-1}\)</p> <p class="math">\(arccsc = csc ^ {-1}\)</p> <br /> - <p class="math">\(deg(rad) = \frac{x}{\frac{\pi}{180}}\)</p> - <p class="math">\(rad(deg) = \frac{x}{\frac{180}{\pi}}\)</p> + <p class="math">\(deg(rad) = \frac{\pi x}{180}\)</p> + <p class="math">\(rad(deg) = \frac{180x}{\pi}\)</p> <br /> <p class="math">\(vinkelsum(x) = \pi(x - 2)\)</p> - <p class="math">\(vinkelsum(3) = \pi(3 - 2) = \pi(1) = \pi\)</p> + <p class="math">\(vinkelsum(3) = \pi(3 - 2) = \pi\)</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> @@ -91,30 +123,30 @@ <p>I en retvinklet trekant:</p> <p class="math">\(\angle C = \frac{\pi}{2}\)</p> <br /> - <p class="math">\(a = c ⋅ sin(\angle A) = c ⋅ cos(\angle B) = b ⋅ tan(\angle A) = b ⋅ cot(\angle B)\)</p> - <p class="math">\(b = c ⋅ sin(\angle B) = c ⋅ cos(\angle A) = a ⋅ tan(\angle B) = a ⋅ cot(\angle A)\)</p> - <p class="math">\(c = a ⋅ csc(\angle A) = b ⋅ csc(\angle B) = a ⋅ sec(\angle B) = b ⋅ sec(\angle A)\)</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>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>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> <!--#include virtual="/include/pgftr.shtml"--> |