On considère l'équation trigonométrique suivante définie sur \(\displaystyle{\mathbb{R}}\) : \(\displaystyle{\cos\left(3x\right) =\dfrac{1}{2}}\) Quelles sont les solutions de cette équation sur \(\displaystyle{\mathbb{R}}\) ? \(\displaystyle{S = \left\{ -\dfrac{\pi}{9}+k\dfrac{2\pi}{3}; \dfrac{\pi}{9}+k\dfrac{2\pi}{3},\ k\in\mathbb{Z} \right\}}\)\(\displaystyle{S = \left\{ -\dfrac{\pi}{9}+k\dfrac{\pi}{3}; \dfrac{\pi}{9}+k\dfrac{\pi}{3},\ k\in\mathbb{Z} \right\}}\)\(\displaystyle{S = \left\{ -\dfrac{\pi}{3}+k\dfrac{2\pi}{3}; \dfrac{\pi}{3}+k\dfrac{2\pi}{3},\ k\in\mathbb{Z} \right\}}\)\(\displaystyle{S = \left\{ -\dfrac{\pi}{9}+k{2\pi}; \dfrac{\pi}{9}+k{2\pi},\ k\in\mathbb{Z} \right\}}\) Quelles sont les solutions de cette équation sur \(\displaystyle{\left[ -\pi ; \pi \right]}\) ? \(\displaystyle{S = \left\{- \dfrac{7\pi}{9} ; - \dfrac{5\pi}{9}; - \dfrac{\pi}{9} ; \dfrac{\pi}{9}; \dfrac{5\pi}{9}; \dfrac{7\pi}{9} \right\}}\)\(\displaystyle{S = \left\{- \dfrac{5\pi}{9}; - \dfrac{\pi}{9} ; \dfrac{\pi}{9}; \dfrac{5\pi}{9}; \right\}}\)\(\displaystyle{S = \left\{ - \dfrac{\pi}{9} ; \dfrac{\pi}{9}\right\}}\)\(\displaystyle{S = \left\{- \dfrac{7\pi}{9} ; - \dfrac{4\pi}{9}; - \dfrac{\pi}{9} ; \dfrac{\pi}{9}; \dfrac{4\pi}{9}; \dfrac{7\pi}{9} \right\}}\)
Quelles sont les solutions de cette équation sur \(\displaystyle{\mathbb{R}}\) ? \(\displaystyle{S = \left\{ -\dfrac{\pi}{9}+k\dfrac{2\pi}{3}; \dfrac{\pi}{9}+k\dfrac{2\pi}{3},\ k\in\mathbb{Z} \right\}}\)\(\displaystyle{S = \left\{ -\dfrac{\pi}{9}+k\dfrac{\pi}{3}; \dfrac{\pi}{9}+k\dfrac{\pi}{3},\ k\in\mathbb{Z} \right\}}\)\(\displaystyle{S = \left\{ -\dfrac{\pi}{3}+k\dfrac{2\pi}{3}; \dfrac{\pi}{3}+k\dfrac{2\pi}{3},\ k\in\mathbb{Z} \right\}}\)\(\displaystyle{S = \left\{ -\dfrac{\pi}{9}+k{2\pi}; \dfrac{\pi}{9}+k{2\pi},\ k\in\mathbb{Z} \right\}}\)
Quelles sont les solutions de cette équation sur \(\displaystyle{\mathbb{R}}\) ? \(\displaystyle{S = \left\{ -\dfrac{\pi}{9}+k\dfrac{2\pi}{3}; \dfrac{\pi}{9}+k\dfrac{2\pi}{3},\ k\in\mathbb{Z} \right\}}\)\(\displaystyle{S = \left\{ -\dfrac{\pi}{9}+k\dfrac{\pi}{3}; \dfrac{\pi}{9}+k\dfrac{\pi}{3},\ k\in\mathbb{Z} \right\}}\)\(\displaystyle{S = \left\{ -\dfrac{\pi}{3}+k\dfrac{2\pi}{3}; \dfrac{\pi}{3}+k\dfrac{2\pi}{3},\ k\in\mathbb{Z} \right\}}\)\(\displaystyle{S = \left\{ -\dfrac{\pi}{9}+k{2\pi}; \dfrac{\pi}{9}+k{2\pi},\ k\in\mathbb{Z} \right\}}\)
\(\displaystyle{S = \left\{ -\dfrac{\pi}{9}+k\dfrac{2\pi}{3}; \dfrac{\pi}{9}+k\dfrac{2\pi}{3},\ k\in\mathbb{Z} \right\}}\)
\(\displaystyle{S = \left\{ -\dfrac{\pi}{9}+k\dfrac{\pi}{3}; \dfrac{\pi}{9}+k\dfrac{\pi}{3},\ k\in\mathbb{Z} \right\}}\)
\(\displaystyle{S = \left\{ -\dfrac{\pi}{3}+k\dfrac{2\pi}{3}; \dfrac{\pi}{3}+k\dfrac{2\pi}{3},\ k\in\mathbb{Z} \right\}}\)
\(\displaystyle{S = \left\{ -\dfrac{\pi}{9}+k{2\pi}; \dfrac{\pi}{9}+k{2\pi},\ k\in\mathbb{Z} \right\}}\)
Quelles sont les solutions de cette équation sur \(\displaystyle{\left[ -\pi ; \pi \right]}\) ? \(\displaystyle{S = \left\{- \dfrac{7\pi}{9} ; - \dfrac{5\pi}{9}; - \dfrac{\pi}{9} ; \dfrac{\pi}{9}; \dfrac{5\pi}{9}; \dfrac{7\pi}{9} \right\}}\)\(\displaystyle{S = \left\{- \dfrac{5\pi}{9}; - \dfrac{\pi}{9} ; \dfrac{\pi}{9}; \dfrac{5\pi}{9}; \right\}}\)\(\displaystyle{S = \left\{ - \dfrac{\pi}{9} ; \dfrac{\pi}{9}\right\}}\)\(\displaystyle{S = \left\{- \dfrac{7\pi}{9} ; - \dfrac{4\pi}{9}; - \dfrac{\pi}{9} ; \dfrac{\pi}{9}; \dfrac{4\pi}{9}; \dfrac{7\pi}{9} \right\}}\)
Quelles sont les solutions de cette équation sur \(\displaystyle{\left[ -\pi ; \pi \right]}\) ? \(\displaystyle{S = \left\{- \dfrac{7\pi}{9} ; - \dfrac{5\pi}{9}; - \dfrac{\pi}{9} ; \dfrac{\pi}{9}; \dfrac{5\pi}{9}; \dfrac{7\pi}{9} \right\}}\)\(\displaystyle{S = \left\{- \dfrac{5\pi}{9}; - \dfrac{\pi}{9} ; \dfrac{\pi}{9}; \dfrac{5\pi}{9}; \right\}}\)\(\displaystyle{S = \left\{ - \dfrac{\pi}{9} ; \dfrac{\pi}{9}\right\}}\)\(\displaystyle{S = \left\{- \dfrac{7\pi}{9} ; - \dfrac{4\pi}{9}; - \dfrac{\pi}{9} ; \dfrac{\pi}{9}; \dfrac{4\pi}{9}; \dfrac{7\pi}{9} \right\}}\)
\(\displaystyle{S = \left\{- \dfrac{7\pi}{9} ; - \dfrac{5\pi}{9}; - \dfrac{\pi}{9} ; \dfrac{\pi}{9}; \dfrac{5\pi}{9}; \dfrac{7\pi}{9} \right\}}\)
\(\displaystyle{S = \left\{- \dfrac{5\pi}{9}; - \dfrac{\pi}{9} ; \dfrac{\pi}{9}; \dfrac{5\pi}{9}; \right\}}\)
\(\displaystyle{S = \left\{- \dfrac{7\pi}{9} ; - \dfrac{4\pi}{9}; - \dfrac{\pi}{9} ; \dfrac{\pi}{9}; \dfrac{4\pi}{9}; \dfrac{7\pi}{9} \right\}}\)