Formulaire de trigonométrie Trigonométrie de base Cercle trigonométrique Similitude cos(θ ) = x P(θ ) tan(θ ) A tan(θ ) = B y sin(θ ) tan(θ ) = = x cos(θ ) 1 Cercle trigonométrique étendu Thm. de Pythagore C2 = A2 + B2 c( θ) C sin(θ ) = y cos2 (θ ) + sin2 (θ ) = 1 sin(θ ) se P(θ ) 1 A Rapports trigonométriques tan(θ ) a y θ x A B C = = a b c a A b B = = etc. c C c c B C b c 1 Dans deux triangles semblables, les proportions des côtés correspondants sont toujours les mêmes. tan(θ ) sin(θ = 1 cos(θ ) sec(θ ) 1 = 1 cos(θ ) 1 + tan2 (θ ) = sec2 (θ ) θ cos(θ ) A Loi des sin cos(θ ) Loi des cos ctg2 (θ ) + 1 = csc2 (θ ) θ B h A A B C = = sin(α) sin(β ) sin(γ) B C γ β ctg(θ ) cos(θ = 1 sin(θ ) csc(θ ) 1 = 1 sin(θ ) Fonctions trigonométriques inverses α C cs c( θ) θ P(θ ) 1 O ctg(θ ) sin(θ ) H O sin(θ ) = H A cos(θ ) = H O tan(θ ) = A θ A C2 = A2 + B2 − 2AB cos(θ ) cos(θ ) = x ⇐⇒ θ = acos(x) θ Triangles comportant des angles usuels sin(θ ) = y ⇐⇒ θ = asin(y) θ π/4 1 √ 2 2 1 π/3 π/6 √ 3/2 1 tan(θ ) = m ⇐⇒ θ = atan(m) 1 1 5π/12 π/12 p √ 2 + 3/2 1 asin(x) ∈ [−π/2, π/2] 1 2 π/4 √ 2 2 acos(x) ∈ [0, π] θ p √ 2− 3 2 atan(x) ∈] − π/2, π/2[ θ ) (ϕ s in Symétries et rotations P(θ ) θ P(π − θ ) P(θ ) ) (ϕ cos cos(−θ ) = cos(θ ) ϕ θ sin(−θ ) = − sin(θ ) sin(θ ) sin(ϕ) cos(π − θ ) = − cos(θ ) θ cos(θ ) cos(ϕ) sin(π − θ ) = sin(θ ) cos(θ ± π) = − cos(θ ) P(θ ± π) P θ + π2 P(θ ± π) P(−θ ) π P 3π 2 −θ P(θ ± π) cos − θ = sin(θ ) π2 sin − θ = cos(θ ) π P 2 −θ 2 3π cos − θ = − sin(θ ) 2 3π − θ = − cos(θ ) sin 2 P(θ ) cos(θ + ϕ) = cos(θ ) cos(ϕ) − sin(θ ) sin(ϕ) sin(θ ± π) = − sin(θ ) π P(θ ) cos θ + = − sin(θ ) 2 π = cos(θ ) sin θ + 2 π cos θ − = sin(θ ) 2 π P θ − π2 = − cos(θ ) sin θ − 2 sin(θ ) cos(ϕ) cos(θ ) sin(ϕ) P(θ + ϕ) sin(θ + ϕ) = sin(θ ) cos(ϕ) + cos(θ ) sin(ϕ) Somme d’angles Identitiés trigonométriques Multiples d’angles sin(2θ ) = 2 sin(θ ) cos(θ ) cos(2θ ) = cos2 (θ ) − sin2 (θ ) Carrés de sinus et cosinus sin2 (θ ) = 1 − cos(2θ ) 2 cos2 (θ ) = 1 + cos(2θ ) 2 Produits de sinus et cosinus sin(θ − ϕ) + sin(θ + ϕ) 2 sin(θ − ϕ) + sin(θ + ϕ) cos(θ ) sin(ϕ) = 2 cos(θ − ϕ) + cos(θ + ϕ) cos(θ ) cos(ϕ) = 2 cos(θ − ϕ) − cos(θ + ϕ) sin(θ ) sin(ϕ) = 2 sin(θ ) cos(ϕ) =