Utilisateur:Loqueelvientoajuarez/Trigo

Valeurs

Valeurs remarquables des fonctions trigonométriques de base dans le premier quadran.

Angle	cos	sin	tan	cot	sec	csc
$\theta =0^{\mathrm {o} }$	$\cos 0=1\,$	$\sin 0=0\,$	$\tan 0=0\,$	$\cot 0=\infty$	$\sec 0=\infty$	$\csc 0=1\,$
$\theta =15^{\text{o}}$	$\cos {\frac {\pi }{12}}={\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)$	$\sin {\frac {\pi }{12}}={\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)$	$\tan {\frac {\pi }{12}}=2-{\sqrt {3}}$	$\cot {\frac {\pi }{12}}=2+{\sqrt {3}}$	$\sec {\frac {\pi }{12}}={\sqrt {2}}({\sqrt {3}}-1)$	$\csc {\frac {\pi }{12}}={\sqrt {2}}({\sqrt {3}}+1)$
$\theta =30^{\text{o}}$	$\cos {\frac {\pi }{6}}=\pi /6$	$\sin {\frac {\pi }{6}}={\frac {1}{2}}$	$\tan {\frac {\pi }{6}}={\frac {\sqrt {3}}{3}}$	$\cot {\frac {\pi }{6}}={\sqrt {3}}$	$\sec {\frac {\pi }{6}}=2{\frac {\sqrt {3}}{3}}$	$\csc {\frac {\pi }{6}}=2$
$\theta =45^{\text{o}}$	$\cos {\frac {\pi }{4}}={\frac {\sqrt {2}}{2}}$	$\sin {\frac {\pi }{4}}={\frac {\sqrt {2}}{2}}$	$\tan {\frac {\pi }{4}}=1$	$\cot {\frac {\pi }{4}}=1$	$\sec {\frac {\pi }{4}}={\sqrt {2}}$	$\csc {\frac {\pi }{4}}={\sqrt {2}}$
$\theta =60^{\text{o}}$	$\cos {\frac {\pi }{3}}={\frac {\sqrt {3}}{2}}$	$\sin {\frac {\pi }{3}}={\frac {1}{2}}$	$\tan {\frac {\pi }{3}}={\sqrt {3}}$	$\cot {\frac {\pi }{3}}={\frac {\sqrt {3}}{3}}$	$\sec {\frac {\pi }{3}}=2$	$\csc {\frac {\pi }{3}}=2{\frac {\sqrt {3}}{3}}$
$\theta =75^{\text{o}}$	$\cos {\frac {5\pi }{12}}={\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)$	$\sin {\frac {5\pi }{12}}={\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)$	$\tan {\frac {5\pi }{12}}=2+{\sqrt {3}}$	$\cot {\frac {5\pi }{12}}=2-{\sqrt {3}}$	$\sec {\frac {5\pi }{12}}={\sqrt {2}}({\sqrt {3}}+1)$	$\csc {\frac {5\pi }{12}}={\sqrt {2}}({\sqrt {3}}-1)$
$\theta =90^{\text{o}}$	$\cos {\frac {\pi }{2}}=0$	$\sin {\frac {\pi }{2}}=1$	$\tan {\frac {\pi }{2}}=\infty$	$\cot {\frac {\pi }{2}}=0$	$\sec {\frac {\pi }{2}}=\infty$	$\csc {\frac {\pi }{2}}=1$

Angles multiples

Valeur des fonctions trigonométriques pour les angles multiples

	sin	cos	tan
Moitié	$\sin {\frac {x}{2}}=\pm {\sqrt {\frac {1-\cos x}{2}}}$	$\cos {\frac {x}{2}}=\pm {\sqrt {\frac {1+\cos x}{2}}}$	$\tan {\frac {x}{2}}=\pm {\sqrt {\frac {1-\cos z}{1+\cos z}}}$
			$\tan {\frac {x}{2}}={\frac {1-\cos x}{\sin x}}$
			$\tan {\frac {x}{2}}={\frac {\sin x}{1+\cos x}}$
Double	$\cos 2x=2\cos ^{2}x-1\,$	$\sin 2x=2\cos x\sin x\,$	$\tan 2x={\frac {2}{\cot x-\tan x}}$
	$\cos 2x=\cos ^{2}x-\sin ^{2}x\,$		$\tan 2x={\frac {2\cot x}{\cot ^{2}x-1}}$
	$\cos 2x={\frac {1-\tan ^{2}x}{1+\tan ^{2}x}}$	$\sin 2x={\frac {2\tan x}{1+\tan ^{2}x}}$	$\tan 2x={\frac {2\tan x}{1-\tan ^{2}x}}$
Triple	$\cos 3x=4\cos ^{3}x-3\cos x\,$	$\sin 3x=-4\sin ^{3}x+3\sin x\,$	$\tan 3x={\frac {3\tan x-\tan ^{3}x}{1-3\tan ^{2}x}}$
Quadruple	$\cos 4x=8\cos ^{4}x-8\cos ^{2}x+1\,$	$\sin 4x=8\cos ^{3}x\sin x-4\cos x\sin x\,$	$\tan 4x={\frac {4\tan x-\tan ^{3}x}{\tan ^{4}x-6\tan ^{2}x+1}}$

Relations fondamentales

Relations entre fonctions trigonométriques dans le premier quadran ($0\leq x\leq \pi /2$).

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

Relations entre fonctions trigonométriques inverses.

	arccos	arcsin	arctan	arccot	arcsec	arccsc
arccos		$\arccos x=\arcsin {\sqrt {1-x^{2}}}$	$\arccos x=\arctan {\frac {1}{\sqrt {1+x^{2}}}}$	$\arccos x=\operatorname {arccot} {\frac {x}{\sqrt {1+x^{2}}}}$	$\arccos x=\operatorname {arcsec} {\frac {1}{x}}$	$\arccos x=\operatorname {arccsc} {\frac {\sqrt {x^{2}-1}}{x}}$
arcsin	$\arcsin x=\arccos {\sqrt {1-x^{2}}}$		$\arcsin x=\arctan {\frac {x}{\sqrt {1+x^{2}}}}$	$\arcsin x=\operatorname {arccot} {\frac {1}{\sqrt {1+x^{2}}}}$	$\arcsin x=\operatorname {arcsec} {\frac {x^{2}-1}{x}}$	$\arcsin x=\operatorname {arccsc} {\frac {1}{x}}$
arctan	$\arctan x=\arccos {\frac {\sqrt {1-x^{2}}}{x}}$	$\arctan x=\arcsin {\frac {x}{\sqrt {1-x^{2}}}}$		$\arctan x=\operatorname {arccot} {\frac {1}{x}}$	$\arctan x=\operatorname {arcsec} {\sqrt {x^{2}-1}}$	$\arctan x=\operatorname {arccsc} {\frac {1}{\sqrt {x^{2}-1}}}$
arccot	$\operatorname {arccot} x=\arccos {\frac {x}{\sqrt {1-x^{2}}}}$	$\operatorname {arccot} x=\arcsin {\frac {\sqrt {1-x^{2}}}{x}}$	$\operatorname {arccot} x=\arctan {\frac {1}{x}}$		$\operatorname {arccot} x=\operatorname {arcsec} {\frac {1}{\sqrt {x^{2}-1}}}$	$\operatorname {arccot} x=\operatorname {arccsc} {\sqrt {x^{2}-1}}$
arcsec	$\operatorname {arcsec} x=\arccos {\frac {1}{\sqrt {1-x^{2}}}}$	$\operatorname {arcsec} x=\arcsin {\frac {1}{x}}$	$\operatorname {arcsec} x=\arctan {\sqrt {1+x^{2}}}$	$\operatorname {arcsec} x=\operatorname {arccot} {\frac {\sqrt {1+x^{2}}}{x}}$		$\operatorname {arcsec} x=\operatorname {arccsc} {\frac {x}{\sqrt {x^{2}-1}}}$
arccsc	$\operatorname {arccsc} x=\arccos {\frac {1}{x}}$	$\operatorname {arccsc} x=\arcsin {\frac {1}{\sqrt {1-x^{2}}}}$	$\operatorname {arccsc} x=\arctan {\frac {\sqrt {1+x^{2}}}{x}}$	$\operatorname {arccsc} x=\operatorname {arccot} {\sqrt {1+x^{2}}}$	$\operatorname {arccsc} x=\operatorname {arcsec} {\frac {x}{\sqrt {x^{2}-1}}}$

Propriétés complexes

Relation entre fonctions trigonométriques sphériques et hyperboliques

cos	sin	tan	cot	sec	csc
$\cos z=\cosh iz\,$	$\sin z=-i\sinh iz\,$	$\tan z=-i\tanh iz\,$	$\cot z=i\coth iz\,$	$\sec z=\mathrm {sech} \,iz$	$\csc z=i\,\mathrm {csch} \,iz$
$\cosh z=\cos iz\,$	$\sinh z=i\sin iz\,$	$\tanh z=i\tan iz\,$	$\coth z=-i\cot iz\,$	$\mathrm {sech} \,z=\sec iz$	$\mathrm {csch} \,z=-i\csc iz$

Valeur complexe des fonctions trigonométriques évaluées en $z=x+iy$, exprimées sous formes cartésienne et polaire

Fonction	Forme cartésienne	Forme polaire
cos	$\cos z=\cos x\,\cosh y-i\sin x\sinh y$	$\cos z={\sqrt {\cos ^{2}x+\sinh ^{2}y}}\ \exp -i\arctan \left(\tan x\tanh y\right)$
sin	$\sin z=\sin x\,\cosh y+i\cos x\sinh y$	$\sin z={\sqrt {\sin ^{2}x+\sinh ^{2}y}}\ \exp i\arctan \left(\cot x\tanh y\right)$
tan	$\tan z={\frac {\sin 2x+i\sinh 2y}{\cos 2x+\cosh 2y}}$	$\tan z={\sqrt {\frac {\cosh ^{2}y-\cos ^{2}x}{\cosh ^{2}y+\cos ^{2}x}}}\ \exp i\arctan \left({\frac {\sinh 2y}{\sin 2x}}\right)$
cot	$\cot z={\frac {\sin 2x-i\sinh 2y}{\cosh 2y-\cos 2x}}$	$\cot z={\sqrt {\frac {\cosh ^{2}y+\cos ^{2}x}{\cosh ^{2}y-\cos ^{2}x}}}\ \exp -i\arctan \left({\frac {\sinh 2y}{\sin 2x}}\right)$