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Question Number 18036 by tawa tawa last updated on 14/Jul/17

$$\mathrm{solve}:$$$$$$$$4\cos \left(\mathrm{x}\right)+2\sin \left(\mathrm{x}\right)=2+\sqrt{3}$$

Commented by ajfour last updated on 14/Jul/17

$$\mathrm{several}\mathrm{solutions}.$$$$\mathrm{L}.\mathrm{H}.\mathrm{S}.=2(2\cos \mathrm{x}+\sin \mathrm{x})$$$$=2\sqrt{5}(\frac{2}{\sqrt{5}}\cos \mathrm{x}+\frac{1}{\sqrt{5}}\sin \mathrm{x})$$$$\mathrm{if}\mathrm{we}\mathrm{let}\mathit{\varphi }={\tan }^{-1}\left(\frac{1}{2}\right)={\cos }^{-1}\left(\frac{2}{\sqrt{5}}\right)$$$$\mathrm{L}.\mathrm{H}.\mathrm{S}.=2\sqrt{5}\cos (\mathrm{x}-\mathit{\varphi })$$$$\mathrm{if}\mathrm{this}\mathrm{has}\mathrm{to}\mathrm{be}\mathrm{equal}\mathrm{to}2+\sqrt{3},$$$$\mathrm{then}2\sqrt{5}\cos (\mathrm{x}-\mathit{\varphi })=2+\sqrt{3}$$$$\cos (\mathrm{x}-\mathit{\varphi })=\frac{2+\sqrt{3}}{2\sqrt{5}}$$$$\mathrm{x}=\mathit{\varphi }+2\mathrm{n}\pi \pm {\cos }^{-1}\left(\frac{2+\sqrt{3}}{2\sqrt{5}}\right)$$$$\mathrm{x}=2\mathrm{n}\pi +{\cos }^{-1}\left(\frac{2}{\sqrt{5}}\right)\pm {\cos }^{-1}\left(\frac{2+\sqrt{3}}{2\sqrt{5}}\right).$$

Commented by tawa tawa last updated on 14/Jul/17

$$\mathrm{please}\mathrm{workings}$$

Commented by ajfour last updated on 14/Jul/17

$$\mathrm{i}\mathrm{have}\mathrm{also}\mathrm{changed}\mathrm{n}\pi \to 2\mathrm{n}\pi$$$$\mathrm{just}\mathrm{now},\mathrm{please}\mathrm{view}\mathrm{again}\mathrm{ms}.\mathrm{tawa}.$$

Commented by tawa tawa last updated on 14/Jul/17

$$\mathrm{Seen}\mathrm{sir}.\mathrm{i}\mathrm{really}\mathrm{appreciate}.\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

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