Leksikon Matematik
Regneregler
\(x + y = z\)
\(z - y = x\)
\(z - x = y\)
\(x y = z\)
\(\frac{z}{y} = x\)
\(\frac{z}{x} = y\)
\(x ^ y = z\)
\(\sqrt[y]{z} = x\)
\(log_{x}(z) = y\)
\(x ^ n = \frac{1}{x ^ {-n}}, x \lt 0\)
\(x ^ 0 = 1\)
\(x ^ {\frac{a}{b}} = \sqrt[b]{x ^ a}\)
\(\frac{x}{y} = x \cdot \frac{1}{y}\)
\(\frac{x}{y} + n = \frac{x + n y}{y}\)
\(\frac{x}{y} + \frac{a}{b} = \frac{x b + a y}{y b}\)
\(\frac{x}{y} n = \frac{x n}{y}\)
\(\frac{x}{y} \frac{a}{b} = \frac{x a}{y b}\)
\(\frac{x}{\frac{a}{b}} = \frac{xb}{a}\)
\(\frac{\frac{x}{y}}{z} = \frac{x}{yz}\)
\(\frac{\frac{x}{y}}{\frac{a}{b}} = \frac{x b}{y a}\)
\(x ^ a x ^ b = x ^ {a + b}\)
\(\frac{x ^ a}{x ^ b} = x ^ {a - b}\)
\(x ^ a y ^ a = (x y) ^ a\)
\(\frac{x ^ a}{y ^ a} = (\frac{x}{y}) ^ a\)
\((x ^ a) ^ b = x ^ {a b}\)
Ligninger
Andengrads:
\(y = a x ^ 2 + b x + c\)
\(x = \frac{-b \pm \sqrt[2]{d}}{2a}\)
\(d = b ^ 2 - 4 a c\)
Funktioner
\(y = f(x)\)
\(x = f ^ {-1}(y)\)
Lineær:
\(f(x) = a x + b\)
\(a = \frac{y_1 - y_0}{x_1 - x_0}\)
\(b = y - ax\)
\(b = f(0) = 0a + b\)
Eksponentiel:
\(f(x) = b a ^ x\)
\(a = \sqrt[x_1 - x_0]{\frac{y_1}{y_{0}}}\)
\(b = \frac{y}{a ^ x}\)
\(b = f(0) = b a ^ 0 = 1b\)
Potens:
\(f(x) = b x ^ a\)
\(a = \frac{log_n(y_1) - log_n(y_0)}{log_n(x_1) - log_n(x_1)}\)
\(b = \frac{y}{x ^ a}\)
\(b = f(1) = b \cdot 1 ^ a = 1b\)
Kvadratisk (andengrads):
\(f(x) = a x ^ 2 + b x + c\)
Trigonometri
\(modliggende_{\angle A} = hosliggende_{\angle B} = a\)
\(hosliggende_{\angle A} = modliggende_{\angle B} = b\)
\(hypotenuse = modliggende_{\angle C} = c\)
\(sin(\theta) = \frac{modliggende}{hypotenuse}\)
\(cos(\theta) = \frac{hosliggende}{hypotenuse}\)
\(tan(\theta) = \frac{modliggende}{hosliggende}\)
\(cot(\theta) = \frac{hosliggende}{modliggende}\)
\(csc(\theta) = \frac{hypotenuse}{modliggende}\)
\(sec(\theta) = \frac{hypotenuse}{hosliggende}\)
\(sin ^ {-1} (\frac{modliggende}{hypotenuse}) = \theta\)
\(cos ^ {-1} (\frac{hosliggende}{hypotenuse}) = \theta\)
\(tan ^ {-1} (\frac{modliggende}{hosliggende}) = \theta\)
\(cot ^ {-1} (\frac{hosliggende}{modliggende}) = \theta\)
\(csc ^ {-1} (\frac{hypotenuse}{modliggende}) = \theta\)
\(sec ^ {-1} (\frac{hypotenuse}{hosliggende}) = \theta\)
Forkortelser:
\(sin = sinus\)
\(cos = cosinus\)
\(tan = tangens\)
\(cot = cotangens\)
\(csc = cosekant\)
\(sec = sekant\)
\(arcsin = sin ^ {-1}\)
\(arccos = cos ^ {-1}\)
\(arctan = tan ^ {-1}\)
\(arccot = cot ^ {-1}\)
\(arcsec = sec ^ {-1}\)
\(arccsc = csc ^ {-1}\)
\(deg(rad) = \frac{\pi x}{180}\)
\(rad(deg) = \frac{180x}{\pi}\)
\(vinkelsum(x) = \pi(x - 2)\)
\(vinkelsum(3) = \pi(3 - 2) = \pi\)
\(\angle A = sin ^ {-1} (\frac{a}{c}) = cos ^ {-1} (\frac{b}{c}) = tan ^ {-1} (\frac{a}{b}) = vinkelsum(3) - \angle B - \angle C\)
\(\angle B = sin ^ {-1} (\frac{b}{c}) = cos ^ {-1} (\frac{a}{c}) = tan ^ {-1} (\frac{b}{a}) = vinkelsum(3) - \angle A - \angle C\)
\(\angle C = vinkelsum(3) - \angle A - \angle B\)
I en regulær trekant:
\(\angle A = \angle B = \angle C\)
I en retvinklet trekant:
\(\angle C = \frac{\pi}{2}\)
\(a = c \cdot sin(\angle A) = c \cdot cos(\angle B) = b \cdot tan(\angle A) = b \cdot cot(\angle B)\)
\(b = c \cdot sin(\angle B) = c \cdot cos(\angle A) = a \cdot tan(\angle B) = a \cdot cot(\angle A)\)
\(c = a \cdot csc(\angle A) = b \cdot csc(\angle B) = a \cdot sec(\angle B) = b \cdot sec(\angle A)\)
I en regulær trekant:
\(a = b = c\)
I en retvinklet trekant:
\(a = \sqrt[2]{c - b ^ 2}\)
\(b = \sqrt[2]{c - a ^ 2}\)
\(c = \sqrt[2]{a ^ 2 + b ^ 2}\)
I en retvinklet trekant, hvori kateterne har samme længde:
\(a = b = \sqrt[2]{\frac{c ^ 2}{2}}\)
\(O = a + b + c\)
\(A = \frac{b h}{2}\)
Mellem to ligedannede trekanter:
\(\angle A_1 = \angle A_0\)
\(\angle B_1 = \angle B_0\)
\(\angle C_1 = \angle C_0\)
\(k = \frac{a_1}{a_0} = \frac{b_1}{b_0} = \frac{c_1}{c_0}\)
\(a_1 = a_0 k\)
\(b_1 = b_0 k\)
\(c_1 = c_0 k\)
\(O_1 = O_0 k\)
\(A_1 = A_0 k ^ 2\)