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Question Number 175547 by daus last updated on 02/Sep/22

Answered by mr W last updated on 02/Sep/22

$$\cos (\frac{\pi }{2}-x)=\cos (x+\frac{\pi }{3})$$$$\Rightarrow 2k\pi +\frac{\pi }{2}-x=x+\frac{\pi }{3}$$$$\Rightarrow x=k\pi +\frac{\pi }{12}$$$$in(-\pi ,\pi ):-\frac{11\pi }{12},\frac{\pi }{12}$$

Commented by Tawa11 last updated on 15/Sep/22

$$\mathrm{Great}\mathrm{sir}$$

Answered by CElcedricjunior last updated on 02/Sep/22

$$\mathit{s}\mathit{i}\mathit{n}\mathit{x}=\mathit{c}\mathit{o}\mathit{s}(\mathit{x}+\frac{\mathit{\pi }}{3})\mathit{f}\mathit{o}\mathit{r}\mathit{x}\in ]-\pi ;\mathit{\pi }]$$$$\mathit{o}\mathit{r}\mathit{s}\mathit{i}\mathit{n}\mathit{x}=\mathit{c}\mathit{o}\mathit{s}(-\mathit{x}+\frac{\mathit{\pi }}{2})$$$$=>\mathit{c}\mathit{o}\mathit{s}(-\mathit{x}+\frac{\mathit{\pi }}{2})=\mathit{c}\mathit{o}\mathit{s}(\mathit{x}+\frac{\mathit{\pi }}{3})$$$$=>\{\begin{array}{l}\frac{\mathit{\pi }}{2}-\mathit{x}=\mathit{x}+\frac{\mathit{\pi }}{3}+2\mathit{k}\mathit{\pi }\\ \frac{\pi }{2}-\mathit{x}=-\mathit{x}-\frac{\mathit{\pi }}{3}+2\mathit{k}\end{array}\mathit{k}\in \mathbb{Z}$$$$=>\{\begin{array}{l}\mathit{x}=\frac{\mathit{\pi }}{12}+\mathit{k}\mathit{\pi }\\ \end{array}$$$${\mathit{S}}_{]-\mathit{\pi };\mathit{\pi }]}=\{-\frac{11\mathit{\pi }}{12};\frac{\mathit{\pi }}{12}\}$$$$$$$$$$$$$$$$$$$$.......lecelebrecedricjunior.......$$$$$$$$$$$$$$$$$$$$$$$$$$

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