Regneregler

\(x+y=z\)

\(z-y=x\)

\(z-x=y\)

\(\sum_{i=a}^b i=a+(a \pm 1) + \cdots +(b \pm 1)+b\)

\(xy=z\)

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

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

\(\prod_{i=a}^b i=a(a \pm 1) \cdots (b \pm 1)b\)

\(n!=\prod_{i=1}^n i,n \gt 0\)

\(0!=1\)

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

\(zy=x\)

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

\(x^y=z\)

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

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

\(x^n=\prod_{i=1}^n x, n \gt 0\)

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

\(x^0=1\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Ligninger

Andengrads:

\(ax^2+bx+c=0\)

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

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

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:

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

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

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

\(f(0)=0\)

\(f(1)=b\)

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:

\(sin=sinus\)

\(cos=cosinus\)

\(tan=tangens\)

\(cot=cotangens\)

\(csc=cosekant\)

\(sec=sekant\)

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

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

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

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

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

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

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

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

\(\sum \theta=(n-2)\pi\)

... hvori n er antallet af vinkler.

\(\sum \theta=(3-2)\pi=\pi\)

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

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

\(\angle C=(\sum \theta)-\angle A-\angle B\)

I en regulær trekant:

\(\angle A=\angle B=\angle C=\frac{\pi}{3}\)

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)	\(\pi\)
tau	\(\tau\)
den gyldne ratio	\(\phi\)

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

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

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

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

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

\(\tau=2\pi \approx \frac{6\ 283\ 185\ 307}{10^9}\)

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