Formules de base
\( (\sin x)' = \cos x \)
\( (\cos x)' = -\sin x \)
\( (\tan x)' = \sec^2 x \)
\( (\sec x)' = \sec x \tan x \)
\( (\csc x)' = -\csc x \cot x \)
\( (\cot x)' = -\csc^2 x \)
Relations utiles
\( \tan x = \frac{\sin x}{\cos x} \)
\( \sec x = \frac{1}{\cos x} \)
\( \csc x = \frac{1}{\sin x} \)
\( \cot x = \frac{\cos x}{\sin x} \)
Identités trigonométriques utiles
\( \sin^2 x+\cos^2 x=1 \)
\( 1+\tan^2 x=\sec^2 x \)
\( 1+\cot^2 x=\csc^2 x \)
Exemple 1 — Retrouver tan à partir de sin et cos
\( \tan x = \frac{\sin x}{\cos x} \)
Règle du quotient :
\( (\tan x)' = \frac{\cos x \cdot \cos x - \sin x(-\sin x)}{\cos^2 x} \)
\( = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} \)
\( = \frac{1}{\cos^2 x} \)
\( (\tan x)' = \sec^2 x \)
Exemple 2 — Retrouver sec à partir de cos
\( \sec x = \frac{1}{\cos x} = (\cos x)^{-1} \)
\( (\sec x)' = -(\cos x)^{-2}(-\sin x) \)
\( = \frac{\sin x}{\cos^2 x} \)
\( = \frac{1}{\cos x} \cdot \frac{\sin x}{\cos x} \)
\( (\sec x)' = \sec x \tan x \)
Exemple 3 — Retrouver csc à partir de sin
\( \csc x = \frac{1}{\sin x} = (\sin x)^{-1} \)
\( (\csc x)' = -(\sin x)^{-2}\cos x \)
\( = -\frac{\cos x}{\sin^2 x} \)
\( = -\csc x \cot x \)
\( (\csc x)' = -\csc x \cot x \)
Exemple 4 — Retrouver cot à partir de cos et sin
\( \cot x = \frac{\cos x}{\sin x} \)
\( (\cot x)' = \frac{-\sin x \cdot \sin x - \cos x \cdot \cos x}{\sin^2 x} \)
\( = -\frac{\sin^2 x + \cos^2 x}{\sin^2 x} \)
\( = -\frac{1}{\sin^2 x} \)
\( (\cot x)' = -\csc^2 x \)
Exemple 5 — composition trigonométrique
\( f(x)=\sin(x^2) \)
\( f'(x)=\cos(x^2)\cdot 2x \)
\( f'(x)=2x\cos(x^2) \)
Exemple 6 — produit trigonométrique
\( f(x)=x\cos x \)
\( f'(x)=1\cdot \cos x + x(-\sin x) \)
\( f'(x)=\cos x - x\sin x \)
Exemple 7 — quotient trigonométrique
\( f(x)=\frac{\sin x}{x} \)
\( f'(x)=\frac{x\cos x - \sin x}{x^2} \)
\( f'(x)=\frac{x\cos x - \sin x}{x^2} \)