Résoudre une inéquation trigonométrique du type cos(x)y Exercice

Dans chacun des cas suivants, résoudre l'inéquation trigonométrique sur \mathbb{R}.

\cos\left(x\right) \geq \dfrac{1}{2}

S=\bigcup_{k\in\mathbb{Z}}\left[-\dfrac{\pi}{3}+k2\pi ; \dfrac{\pi}{3}+k2\pi \right]

S=\left[-\dfrac{\pi}{3}+k2\pi ; \dfrac{\pi}{3}+k2\pi \right]

S=\bigcup_{k\in\mathbb{Z}}\left[-\dfrac{2\pi}{3}+k2\pi ; \dfrac{2\pi}{3}+k2\pi \right]

S=\left[-\dfrac{2\pi}{3}+k2\pi ; \dfrac{2\pi}{3}+k2\pi \right]

\cos\left(x\right) \leq \dfrac{\sqrt3}{2}

S=\bigcup_{k\in\mathbb{Z}}\left[ \dfrac{\pi}{6}+ k2\pi ;\dfrac{11\pi}{6}+ k2\pi\right]

S=\left[ \dfrac{\pi}{6}+ k2\pi ;\dfrac{11\pi}{6}+ k2\pi\right]

S =\bigcup_{k\in\mathbb{Z}}\left[ -\dfrac{\pi}{6}+ k2\pi ;\dfrac{5\pi}{6}+ k2\pi\right]

S=\left[ \dfrac{-\pi}{6}+ k2\pi ;\dfrac{5\pi}{6}+ k2\pi\right]

\cos\left(x\right) \geq \dfrac{\sqrt2}{2}

S= \bigcup_{k\in\mathbb{Z}}\left[-\dfrac{\pi}{4}+k2\pi; \dfrac{\pi}{4}+k2\pi \right]

S = \left[-\dfrac{\pi}{4}; \dfrac{\pi}{4} \right]

S = \bigcup_{k\in\mathbb{Z}}\left[-\dfrac{3\pi}{4}+k2\pi; \dfrac{3\pi}{4}+k2\pi \right]

S = \left[-\dfrac{3\pi}{4}+k2\pi; \dfrac{3\pi}{4}+k2\pi \right]

\cos\left(x\right) \gt 0

S=\bigcup_{k\in\mathbb{Z}} \left]-\dfrac{\pi}{2}+k2\pi ; \dfrac{\pi}{2}+k2\pi \right[

S = \left]-\dfrac{\pi}{2} ; \dfrac{\pi}{2} \right[

S =\bigcup_{k\in\mathbb{Z}} \left]-{\pi}+k2\pi ; {\pi}+k2\pi \right[

S = \left]-{\pi} ; {\pi} \right[

\cos\left(x\right) \leq -\dfrac{\sqrt2}{2}

S =\bigcup_{k\in\mathbb{Z}} \left[\dfrac{3\pi}{4}+ k2\pi ; \dfrac{5\pi}{4} + k2\pi\right]

S =\bigcup_{k\in\mathbb{Z}} \left[\dfrac{3\pi}{2}+ k2\pi ; \dfrac{5\pi}{2} + k2\pi\right]

S = \left[\dfrac{3\pi}{4} ; \dfrac{5\pi}{4} \right]

S = \left[\dfrac{\pi}{4} ; \dfrac{3\pi}{4} \right]

\cos\left(x\right) \geq -\dfrac{\sqrt3}{2}

S = \bigcup_{k\in\mathbb{Z}}\left[-\dfrac{5\pi}{6}+k2\pi ; \dfrac{5\pi}{6}+k2\pi \right]

S =\left[-\dfrac{5\pi}{6} ; \dfrac{5\pi}{6} \right]

S = \bigcup_{k\in\mathbb{Z}}\left[-\dfrac{\pi}{6}+k2\pi ; \dfrac{\pi}{6}+k2\pi \right]

S =\left[-\dfrac{\pi}{6} ; \dfrac{\pi}{6} \right]

\cos\left(x\right) \lt -\dfrac{1}{2}

S =\bigcup_{k\in\mathbb{Z}} \left] \dfrac{2\pi}{3}+ k2\pi ; \dfrac{4\pi}{3}+k2\pi \right[

S =\bigcup_{k\in\mathbb{Z}} \left]- \dfrac{2\pi}{3}+ k2\pi ; \dfrac{2\pi}{3}+k2\pi \right[

S = \left] -\dfrac{2\pi}{3}; \dfrac{2\pi}{3} \right[

S = \left] -\dfrac{\pi}{3}; \dfrac{\pi}{3} \right[

\cos\left(x\right) \geq 1

S=\left\{0 + k2\pi, k\in\mathbb{Z}\right\}

S=\left\{k\pi, k\in\mathbb{Z}\right\}

S=\left\{ 0,\pi \right\}

S =\left[ 0,\pi \right]