Regneregler

\(x+y=z\)

\(z-y=x\)

\(z-x=y\)

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

\(xy=z\)

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

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

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

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

\(0!=1\)

\(\frac xy=z\)

\(zy=x\)

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

\(x^y=z\)

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

\(log_x(z)=y\)

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

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

\(x^0=1\)

\(x^{\frac nm}=\sqrt[m] {x^n}\)

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

\(\frac xy+\frac ab=\frac {x b+ay}{yb}\)

\(\frac xyn=\frac {xn}{y}\)

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

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

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

\(\frac {\frac xy}{\frac ab}=\frac {xb}{ya}\)

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

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

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

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

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

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

\(modliggende_{\alpha}=hosliggende_{\beta}=a\)

\(hosliggende_{\alpha}=modliggende_{\beta}=b\)

\(hypotenuse=modliggende_{\gamma}=c\)

\(sin(\theta)=\frac {modliggende_{\theta}}{hypotenuse_{\theta}}\)

\(cos(\theta)=\frac {hosliggende_{\theta}}{hypotenuse_{\theta}}\)

\(tan(\theta)=\frac {modliggende_{\theta}}{hosliggende_{\theta}}\)

\(cot(\theta)=\frac {hosliggende_{\theta}}{modliggende_{\theta}}\)

\(csc(\theta)=\frac {hypotenuse_{\theta}}{modliggende_{\theta}}\)

\(sec(\theta)=\frac {hypotenuse_{\theta}}{hosliggende_{\theta}}\)

\(sin^{-1}(\frac {modliggende_{\theta}}{hypotenuse_{\theta}})=\theta\)

\(cos^{-1}(\frac {hosliggende_{\theta}}{hypotenuse_{\theta}})=\theta\)

\(tan^{-1}(\frac {modliggende_{\theta}}{hosliggende_{\theta}})=\theta\)

\(cot^{-1}(\frac {hosliggende_{\theta}}{modliggende_{\theta}})=\theta\)

\(csc^{-1}(\frac {hypotenuse_{\theta}}{modliggende_{\theta}})=\theta\)

\(sec^{-1}(\frac {hypotenuse_{\theta}}{hosliggende_{\theta}})=\theta\)

Forkortelser:

\(sin=sinus\)

\(cos=cosinus\)

\(tan=tangens\)

\(cot=cotangens\)

\(csc=cosekant\)

\(sec=sekant\)

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

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

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

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

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

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

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

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

\(\Theta(n)=(n-2)\pi\)

hvori \({\Theta}(n)\) er vinkelsummen af n-gonen.

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

\(\alpha=sin^{-1}(\frac ac)=cos^{-1}(\frac bc)=tan^{-1}(\frac ab)=\Theta(3)-\beta-\gamma\)

\(\beta=sin^{-1}(\frac bc)=cos^{-1}(\frac ac)=tan^{-1}(\frac ba)=\Theta(3)-\alpha-\gamma\)

\(\gamma=\Theta(3)-\alpha-\beta\)

I en retvinklet trekant:

\(\gamma=\frac {\pi}2\)

I en regulær trekant:

\(\alpha=\beta=\gamma=\frac {\pi}3\)

\(a=c \cdot sin(\alpha)=c \cdot cos(\beta)=b \cdot tan(\alpha)=b \cdot cot(\beta)\)

\(b=c \cdot sin(\beta)=c \cdot cos(\alpha)=a \cdot tan(\beta)=a \cdot cot(\alpha)\)

\(c=a \cdot csc(\alpha)=b \cdot csc(\beta)=a \cdot sec(\beta)=b \cdot sec(\alpha)\)

I en retvinklet trekant:

\(a=\sqrt[2] {c^2-b^2}\)

\(b=\sqrt[2] {c^2-a^2}\)

\(c=\sqrt[2] {a^2+b^2}\)

I en retvinklet trekant, hvori kateterne har samme længde:

\(a=b=\sqrt[2] {\frac {c^2}2}\)

I en regulær trekant:

\(a=b=c\)

\(O=a+b+c\)

\(A=\frac {bh}2\)

hvori b er bredten af trekanten og h er dens højde.

Mellem to ligedannede trekanter:

\(\alpha_1=\alpha_0\)

\(\beta_1=\beta_0\)

\(\gamma_1=\gamma_0\)

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

\(a_1=a_0 k\)

\(b_1=b_0 k\)

\(c_1=c_0 k\)

\(O_1=O_0 k\)

\(A_1=A_0 k^2\)

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	\(\phi\)

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

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

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

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

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

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

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