Série 2 — formules d'addition, duplication et équations.
Exercice 1
Calculer les valeurs exactes de \(\cos\dfrac{7\pi}{12}\) et \(\sin\dfrac{7\pi}{12}\).
Correction
\(\dfrac{7\pi}{12} = \dfrac{\pi}{3} + \dfrac{\pi}{4}\). \(\cos = \cos\dfrac{\pi}{3}\cos\dfrac{\pi}{4} - \sin\dfrac{\pi}{3}\sin\dfrac{\pi}{4} = \dfrac{\sqrt2 - \sqrt6}{4}\). \(\sin = \sin\dfrac{\pi}{3}\cos\dfrac{\pi}{4} + \cos\dfrac{\pi}{3}\sin\dfrac{\pi}{4} = \dfrac{\sqrt6 + \sqrt2}{4}\).
Exercice 2
Sachant que \(\cos x = \dfrac35\) avec \(x\in\left]0\,;\dfrac{\pi}{2}\right[\), calculer \(\cos 2x\) et \(\sin 2x\).
Correction
\(\sin x = \dfrac45\) ; \(\cos 2x = 2\cos^2x - 1 = \dfrac{18}{25} - 1 = -\dfrac{7}{25}\) ; \(\sin 2x = 2\sin x\cos x = \dfrac{24}{25}\).
Exercice 3
Résoudre dans \([0\,;2\pi[\) : \(2\cos^2 x - 3\cos x + 1 = 0\).
Correction
On pose \(X = \cos x\) : \(2X^2 - 3X + 1 = 0\) donne \(X = 1\) ou \(X = \dfrac12\). \(\cos x = 1\) : \(x = 0\). \(\cos x = \dfrac12\) : \(x = \dfrac{\pi}{3}\) ou \(x = \dfrac{5\pi}{3}\).
Exercice 4
Montrer que \(\cos^4 x - \sin^4 x = \cos 2x\).
Correction
\(\cos^4 x - \sin^4 x = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) = \cos 2x\times1 = \cos 2x\).
Exercice 5
Résoudre dans \(\mathbb{R}\) : \(\sin 2x = \sin x\).
Correction
\(2\sin x\cos x - \sin x = 0\), soit \(\sin x(2\cos x - 1) = 0\). \(\sin x = 0\) donne \(x = k\pi\) ; \(\cos x = \dfrac12\) donne \(x = \pm\dfrac{\pi}{3} + 2k\pi\), avec \(k\in\mathbb{Z}\).
Exercice 6
Simplifier \(A = \dfrac{\sin 2x}{1 + \cos 2x}\) pour \(\cos x \ne 0\).
Correction
\(\sin 2x = 2\sin x\cos x\) et \(1 + \cos 2x = 2\cos^2 x\), donc \(A = \dfrac{2\sin x\cos x}{2\cos^2 x} = \tan x\).
Exercice 7
Résoudre dans \([0\,;2\pi[\) : \(\cos\left(2x - \dfrac{\pi}{3}\right) = \dfrac{\sqrt3}{2}\).
Correction
\(2x - \dfrac{\pi}{3} = \pm\dfrac{\pi}{6} + 2k\pi\), donc \(2x = \dfrac{\pi}{2} + 2k\pi\) ou \(2x = \dfrac{\pi}{6} + 2k\pi\). Ainsi \(x = \dfrac{\pi}{4} + k\pi\) ou \(x = \dfrac{\pi}{12} + k\pi\). Sur \([0\,;2\pi[\) : \(x\in\left\{\dfrac{\pi}{12}\,;\dfrac{\pi}{4}\,;\dfrac{13\pi}{12}\,;\dfrac{5\pi}{4}\right\}\).