Formulaire de Trigonométrie Angles associés Une lecture efficace du cercle trigonométrique permet de retrouver les relations suivantes : ´ ´ ³π ³π + x = − sin(x) − x = sin(x) cos cos π π ³ π2 ³ π2 ´ ´ +x −x 2 2 sin sin + x = cos(x) − x = cos(x) 2 2 cos (π − x) = − cos(x) sin (π − x) = sin(x) π−x x cos (π + x) = − cos(x) sin (π + x) = − sin(x) π+x −x cos (−x) = cos(x) sin (−x) = − sin(x) Valeurs remarquables 0 cos sin tan 1 0 0 π 6 p 3 2 π 4 p 2 2 p 2 2 1 2 p 3 3 1 π 3 π 2 π 1 2 p 3 2 0 −1 1 0 p 3 0 Relations entre cos, sin et tan cos2 (x) + sin2 (x) = 1 1 + tan2 (x) = 1 cos2 (x) si x 6≡ π [π] 2 Formules d’addition cos(a + b) = cos(a) cos(b) − sin(a) sin(b) sin(a + b) = sin(a) cos(b) + sin(b) cos(a) cos(a − b) = cos(a) cos(b) + sin(a) sin(b) sin(a − b) = sin(a) cos(b) − sin(b) cos(a) tan(a) + tan(b) tan(a + b) = 1 − tan(a) tan(b) tan(a − b) = tan(a) − tan(b) 1 + tan(a) tan(b) Formules de duplication cos(2a) = cos2 (a) − sin2 (a) = 2cos2 (a) − 1 sin(2a) = 2sin(a) cos(a) tan(2a) = 2tan(a) 1 − tan2 (a) = 1 − 2sin2 (a) Transformation de somme en produit ³p +q´ ³p −q´ cos(p) + cos(q) = 2cos cos 2 2 ³p −q´ ³p +q ´ cos sin(p) + sin(q) = 2sin 2 2 ³p +q ´ ³p −q´ cos(p) − cos(q) = −2sin sin 2 2 ³p −q ´ ³p +q ´ sin(p) − sin(q) = 2sin cos 2 2 Transformation de produit en somme 1 [sin(a + b) + sin(a − b)] 2 1 cos(a) cos(b) = [cos(a + b) + cos(a − b)] 2 1 sin(a) sin(b) = − [cos(a + b) − cos(a − b)] 2 sin(a) cos(b) = cos2 (a) = 1 + cos(2a) 2 sin2 (a) = cos, sin et tan en fonction de l’angle moitié ³a´ 1− t2 2t Si t = tan , on a : cos(a) = ; sin(a) = 2 1+ t2 1+ t2 Equations trigonométriques cos(a) = cos(b) ⇐⇒ ½ a b = −a a ≡ b[2π] a ≡ −b[2π] 1 − cos(2a) 2 ; tan(a) = sin(a) = sin(b) ⇐⇒ b = π−a ½ tan2 (a) = 2t 1− t2 a ≡ b[2π] a ≡ π − b[2π] a 1 − cos(2a) 1 + cos(2a) tan(a) = tan(b) ⇐⇒ a ≡ b[π]