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Si \(\alpha\) est un angle du troisième quadrant tel que \(\sin\alpha\cdot\cos\alpha=\dfrac{1}{2}\), calculez \(\sin\alpha+\cos\alpha\) .
\( \sqrt{2} \)
\( -\sqrt{2} \)
\( -1 \)
impossible
Résolvez l'équation \(\sin x = \cos x \).
\( S=\emptyset \)
\( S=\left\{\dfrac{\pi}{4}+k\pi;\, k\in\mathbb{Z}\right\} \)
\(S=\left\{\dfrac{\pi}{4}+2k\pi;\, k\in\mathbb{Z}\right\} \)
\( S=\left\{\dfrac{\pi}{6}+2k\pi,\, \dfrac{5\pi}{6}+2k\pi;\, k\in\mathbb{Z}\right\} \)
Résolvez l'équation \(4\cos^4 x-5\cos^2 x+1 =0\) .
\( S=\left\{\dfrac{1}{4}+2k\pi,\, 1+2k\pi;\, k\in\mathbb{Z}\right\}\)
\( S=\left\{-\dfrac{\pi}{3}+2k\pi,\, \dfrac{\pi}{3}+2k\pi,\, 2k\pi;\, k\in\mathbb{Z}\right\} \)
\( S=\left\{-\dfrac{2\pi}{3}+2k\pi,\, -\dfrac{\pi}{3}+2k\pi,\, \dfrac{\pi}{3}+2k\pi,\, \dfrac{2\pi}{3}+2k\pi,\, \pi+2k\pi,\, 2k\pi;\, k\in\mathbb{Z}\right\} \)
\(S=\emptyset \)
A l'aide des formules, calculez \(\cos\left(\dfrac{7\pi}{12}\right)\) .
\( \dfrac{\sqrt{2}-\sqrt{6}}{4} \)
\( \dfrac{\sqrt{2}+\sqrt{6}}{4} \)
\( \dfrac{\sqrt{6}-\sqrt{2}}{4} \)
\( \dfrac{1+\sqrt{2}}{2} \)
Résolvez l'équation \(\sin^4 x+\sin^2 x-2 =0 \).
\( S=\left\{-2+2k\pi,\, 1+2k\pi;\, k\in\mathbb{Z}\right\} \)
\( S=\left\{\dfrac{\pi}{2}+2k\pi;\, k\in\mathbb{Z}\right\} \)
\( S=\left\{\dfrac{\pi}{2}+2k\pi,\, \dfrac{3\pi}{2}+2k\pi;\, k\in\mathbb{Z}\right\} \)
A l'aide des formules, calculez \(\cos\left(\dfrac{5\pi}{12}\right)\) .
\( \dfrac{\sqrt{6}+\sqrt{2}}{4} \)
\( \dfrac{\sqrt{3}+\sqrt{2}}{2} \)
Résolvez l'équation \( tg^2 x-3tg\, x+2 =0\) .
\( S=\left\{1+k\pi,\, 2+k\pi;\, k\in\mathbb{Z}\right\} \)
\( S=\left\{\dfrac{\pi}{4}+k\pi,\, 1,107+k\pi;\, k\in\mathbb{Z}\right\} \)
Résolvez l'équation \(2\sin^2{x}=1-\sin{x} \).
\( S=\left\{-1,\, \dfrac{1}{2}\right\} \)
\( S=\left\{\dfrac{\pi}{6},\, \dfrac{5\pi}{6},\, \dfrac{3\pi}{2}\right\} \)
\( S=\left\{\dfrac{\pi}{6}+2k\pi,\, \dfrac{3\pi}{2}+2k\pi;\, k\in\mathbb{Z}\right\} \)
\( S=\left\{\dfrac{\pi}{6}+2k\pi,\, \dfrac{5\pi}{6}+2k\pi,\, \dfrac{3\pi}{2}+2k\pi;\, k\in\mathbb{Z}\right\} \)
Résolvez l'équation \(2\cos^2 x-3\cos x+1 =0\) .
\( S=\left\{1+2k\pi,\, \dfrac{1}{2}+2k\pi;\, k\in\mathbb{Z}\right\} \)
\( S=\left\{2k\pi;\, k\in\mathbb{Z}\right\} \)
\( S=\left\{\dfrac{\pi}{3}+2k\pi,\, 2k\pi;\, k\in\mathbb{Z}\right\} \)
\( S=\left\{\dfrac{\pi}{3}+2k\pi,\, -\dfrac{\pi}{3}+2k\pi,\, 2k\pi;\, k\in\mathbb{Z}\right\} \)
Résolvez l'équation \(\sin{\dfrac{x}{2}}+\cos{x}=1\) .
\( S=\left\{2k\pi,\, \dfrac{\pi}{3}+4k\pi,\, \dfrac{5\pi}{3}+4k\pi;\, k\in\mathbb{Z}\right\} \)
\( S=\left\{2k\pi,\, \dfrac{\pi}{3}+4k\pi;\, k\in\mathbb{Z}\right\} \)
\( S=\left\{0,\, \dfrac{\pi}{3},\, \dfrac{5\pi}{3}\right\} \)
\( S=\left\{2k\pi,\, \dfrac{\pi}{3}+2k\pi,\, \dfrac{5\pi}{3}+2k\pi;\, k\in\mathbb{Z}\right\} \)
