Regneregler

\(x + y = z\)

\(z - y = x\)

\(z - x = y\)

\(x y = z\)

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

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

\(x ^ {y} = z\)

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

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

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

\(x ^ {0} = 1\)

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

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

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

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

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

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

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

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

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

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

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

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

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

Trigonometri

\(modliggende_{\angle A} = hosliggende_{\angle B} = a\)

\(hosliggende_{\angle A} = modliggende_{\angle B} = b\)

\(hypotenuse = modliggende_{\angle C} = c\)

Forkortelser:

\(sin_{sinus}\)

\(cos_{cosinus}\)

\(tan_{tangens}\)

\(cot_{cotangens}\)

\(csc_{cosekant}\)

\(sec_{sekant}\)

\(sin(\theta) = \frac{modliggende}{hypotenuse}\)

\(cos(\theta) = \frac{hosliggende}{hypotenuse}\)

\(tan(\theta) = \frac{modliggende}{hosliggende}\)

\(cot(\theta) = \frac{hosliggende}{modliggende}\)

\(csc(\theta) = \frac{hypotenuse}{modliggende}\)

\(sec(\theta) = \frac{hypotenuse}{hosliggende}\)

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

\(sin ^ {-1} (\frac{modliggende}{hypotenuse}) = \theta\)

\(cos ^ {-1} (\frac{hosliggende}{hypotenuse}) = \theta\)

\(tan ^ {-1} (\frac{modliggende}{hosliggende}) = \theta\)

\(cot ^ {-1} (\frac{hosliggende}{modliggende}) = \theta\)

\(csc ^ {-1} (\frac{hypotenuse}{modliggende}) = \theta\)

\(sec ^ {-1} (\frac{hypotenuse}{hosliggende}) = \theta\)

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

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

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

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

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

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

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

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

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

\(vinkelsum(3) = \pi(3 - 2) = \pi(1) = \pi\)

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

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

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

I en regulær trekant:

\(\angle A = \angle B = \angle C\)

I en retvinklet trekant:

\(\angle C = \frac{\pi}{2}\)

\(a = c ⋅ sin(\angle A) = c ⋅ cos(\angle B) = b ⋅ tan(\angle A) = b ⋅ cot(\angle B)\)

\(b = c ⋅ sin(\angle B) = c ⋅ cos(\angle A) = a ⋅ tan(\angle B) = a ⋅ cot(\angle A)\)

\(c = a ⋅ csc(\angle A) = b ⋅ csc(\angle B) = a ⋅ sec(\angle B) = b ⋅ sec(\angle A)\)

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:

\(\angle A_{1} = \angle A_{0}\)

\(\angle B_{1} = \angle B_{0}\)

\(\angle C_{1} = \angle C_{0}\)

\(k = \frac{a_{1}}{a_{0}} = \frac{b_{1}}{b_{0}} = \frac{c_{1}}{c_{0}}\)

\(a_{1} = a_{0} k\)

\(b_{1} = b_{0} k\)

\(c_{1} = c_{0} k\)

\(O_{1} = O_{0} k\)

\(A_{1} = A_{0} k ^ {2}\)