Regneregler

\(x + y = z\)

\(z - y = x\)

\(z - x = y\)

\(xy = z\)

\(\frac{z}{y} = x\)

\(\frac{z}{x} = y\)

\(x ^ y = z\)

\(\sqrt[y]{z} = x\)

\(log_{x}(z) = y\)

\(x ^ n = \frac{1}{x ^ {-n}}, x \lt 0\)

\(x ^ 0 = 1\)

\(x ^ {\frac{a}{b}} = \sqrt[b]{x ^ a}\)

\(\frac{x}{y} = x\frac{1}{y}\)

\(\frac{x}{y} + n = \frac{x + n y}{y}\)

\(\frac{x}{y} + \frac{a}{b} = \frac{x b + ay}{yb}\)

\(\frac{x}{y}n = \frac{xn}{y}\)

\(\frac{x}{y}\frac{a}{b} = \frac{x a}{y b}\)

\(\frac{x}{\frac{a}{b}} = \frac{xb}{a}\)

\(\frac{\frac{x}{y}}{z} = \frac{x}{yz}\)

\(\frac{\frac{x}{y}}{\frac{a}{b}} = \frac{xb}{ya}\)

\(x ^ ax ^ b = x ^ {a + b}\)

\(\frac{x ^ a}{x ^ b} = x ^ {a - b}\)

\(x ^ ay ^ a = (xy) ^ a\)

\(\frac{x ^ a}{y ^ a} = (\frac{x}{y}) ^ a\)

\((x ^ a) ^ b = x ^ {ab}\)

Ligninger

Andengrads:

\(y = ax ^ 2 + bx + c\)

\(x = \frac{-b \pm \sqrt[2]{d}}{2a}\)

\(d = b ^ 2 - 4ac\)

Funktioner

\(y = f(x)\)

\(x = f ^ {-1}(y)\)

Lineær:

\(f(x) = ax + b\)

\(a = \frac{y_1 - y_0}{x_1 - x_0}\)

\(b = y - ax\)

\(f(0) = b\)

Eksponentiel:

\(f(x) = ba ^ x\)

\(a = \sqrt[x_1 - x_0]{\frac{y_1}{y_{0}}}\)

\(b = \frac{y}{a ^ x}\)

\(f(0) = b\)

Potens:

\(f(x) = bx ^ a\)

\(a = \frac{log_n(y_1) - log_n(y_0)}{log_n(x_1) - log_n(x_1)}\)

\(b = \frac{y}{x ^ a}\)

\(f(0) = 0\)

\(f(1) = b\)

Kvadratisk (andengrads):

\(f(x) = ax ^ 2 + bx + c\)

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:

\(sin = sinus\)

\(cos = cosinus\)

\(tan = tangens\)

\(cot = cotangens\)

\(csc = cosekant\)

\(sec = sekant\)

\(arcsin = sin ^ {-1}\)

\(arccos = cos ^ {-1}\)

\(arctan = tan ^ {-1}\)

\(arccot = cot ^ {-1}\)

\(arcsec = sec ^ {-1}\)

\(arccsc = csc ^ {-1}\)

\(deg(rad) = \frac{rad\pi}{180}\)

\(rad(deg) = \frac{deg180}{\pi}\)

\(vinkelsum(x) = (x - 2)\pi\)

\(vinkelsum(3) = (3 - 2)\pi = \pi\)

\(\angle A = sin ^ {-1} (\frac{a}{c}) = cos ^ {-1} (\frac{b}{c}) = tan ^ {-1} (\frac{a}{b}) = vinkelsum(3) - \angle B - \angle C\)

\(\angle B = sin ^ {-1} (\frac{b}{c}) = cos ^ {-1} (\frac{a}{c}) = tan ^ {-1} (\frac{b}{a}) = vinkelsum(3) - \angle A - \angle C\)

\(\angle C = vinkelsum(3) - \angle A - \angle B\)

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:

\(\angle A_1 = \angle A_0\)

\(\angle B_1 = \angle B_0\)

\(\angle C_1 = \angle C_0\)

\(k = \frac{a_1}{a_0} = \frac{b_1}{b_0} = \frac{c_1}{c_0}\)

\(a_1 = a_0 k\)

\(b_1 = b_0 k\)

\(c_1 = c_0 k\)

\(O_1 = O_0 k\)

\(A_1 = A_0 k ^ 2\)

Konstanter

Navn	Symbol
Pythagoras' konstant	\(\sqrt[2]{2}\)
Theodorus' konstant	\(\sqrt[2]{3}\)
Eulers tal	\(e\)
den imaginære enhed	\(i\)
Arkimedes' konstant (pi)	\(\pi\)
den gyldne ratio	\(\phi\)

\(\sqrt[2]{2} \approx \frac{1\ 414\ 213\ 562}{10 ^ 9}\)

\(\sqrt[2]{3} \approx \frac{1\ 732\ 050\ 808}{10 ^ 9}\)

\(e = \sum_{n = 0} ^ \infty \frac{1}{n!} \approx \frac{2\ 718\ 281\ 828}{10 ^ 9}\)

\(i = \sqrt[2]{-1}\)

\(\pi \approx \frac{3\ 141\ 592\ 654}{10 ^ 9}\)

\(\phi = \frac{1 + \sqrt[2]{5}}{2} \approx \frac{1\ 618\ 033\ 989}{10 ^ 9}\)