Goniometrie

Laes hie wie v'r mit de versjillende saorte Limburgs ómgaon.

De goniometrie of trigonometrie is 'nen tak van de wiskónde dae zich besjeftig mit drieheuk en in 't biezunjert de oarsprunkelik op drieheuk gegrundje fónksjes wie sinus (sin), cosinus (cos) en tangens (tan) en de väöl minder-gebroeke respektievelike reciproake fónksjes, secans (sec), cosecans (csc) en cotangens (cot)

${\displaystyle \sec(x)={\frac {1}{\sin x}}}$
${\displaystyle \csc(x)={\frac {1}{\cos x}}}$
${\displaystyle \cot(x)={\frac {1}{\tan x}}}$

De tangens van 'nen hook is gediffenieerdj es de verhaojing tösse de twiè rechte en euverligkendje kantje van 'ne rechheukigen driehook en dus op d'n einheidskrink:

${\displaystyle \tan x={\frac {\sin x}{\cos x}}}$

Volgens Pythagoras geldj op d'n einheidskrink:

${\displaystyle \sin ^{2}x+\cos ^{2}x=1}$

Mit dees gróndjvergeliekinge kan ummer ein fónksje in de twiè anger waeren oetgedrök:

${\displaystyle \sin x={\sqrt {1-\cos ^{2}x}}}$
${\displaystyle \sin x={\frac {\tan x}{\sqrt {1+\tan ^{2}x}}}}$

${\displaystyle \cos x={\sqrt {1-\sin ^{2}x}}}$
${\displaystyle \cos x={\frac {1}{\sqrt {1+\tan ^{2}x}}}}$

${\displaystyle \tan x={\frac {\sqrt {1-\cos ^{2}x}}{\cos x}}}$
${\displaystyle \tan x={\frac {\sin x}{\sqrt {1-\sin ^{2}x}}}}$

De trigonometrische fónksjes höbbe symmetrie:

${\displaystyle \sin(-x)=-\sin x}$
${\displaystyle \cos(-x)=\cos x}$
${\displaystyle \tan(-x)=-\tan x}$

Wiejer zint de völgendje raekenregele van toepassing:

${\displaystyle \sin(x+y)=\sin x\;\cos y+\sin y\;\cos x}$
${\displaystyle \cos(x+y)=\cos y\;\cos x-\sin x\;\sin y}$
${\displaystyle \tan(x+y)={\frac {\tan x+\tan y}{1-\tan x\;\tan y}}}$

Mit x=y levertj det den de völgendje oetdrökkinge op:

${\displaystyle \sin(2\;x)=2\sin x\;\cos x}$
${\displaystyle \cos(2\;x)=\cos ^{2}x-\sin ^{2}x}$
${\displaystyle \tan(2\;x)={\frac {2\tan x}{1-\tan ^{2}x}}}$