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Question

# Prove that $\left(2\sqrt{3}+3\right)\mathrm{sin}x+2\sqrt{3}\mathrm{cos}x$ lies between $-\left(2\sqrt{3}+\sqrt{15}\right)\mathrm{and}\left(2\sqrt{3}+\sqrt{15}\right)$.

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Solution

## $\mathrm{Let}f\left(x\right)=\left(2\sqrt{3}+3\right)\mathrm{sin}x+2\sqrt{3}\mathrm{cos}x\phantom{\rule{0ex}{0ex}}\mathrm{We}\mathrm{know}\mathrm{that},\phantom{\rule{0ex}{0ex}}-\sqrt{{\left(2\sqrt{3}+3\right)}^{2}+{\left(2\sqrt{3}\right)}^{2}}\le f\left(x\right)\le \sqrt{{\left(2\sqrt{3}+3\right)}^{2}+{\left(2\sqrt{3}\right)}^{2}}\phantom{\rule{0ex}{0ex}}⇒-\sqrt{12+9+12\sqrt{3}+12}\le f\left(x\right)\le \sqrt{12+9+12\sqrt{3}+12}\phantom{\rule{0ex}{0ex}}⇒-\sqrt{33+12\sqrt{3}}\le f\left(x\right)\le \sqrt{33+12\sqrt{3}}\phantom{\rule{0ex}{0ex}}\mathrm{Disclaimer}:\mathrm{Instead}\mathrm{of}-\left(2\sqrt{3}+\sqrt{15}\right)\mathrm{and}\left(2\sqrt{3}+\sqrt{15}\right),\mathrm{it}\mathrm{should}\mathrm{be}-\sqrt{33+12\sqrt{3}}\mathrm{and}\sqrt{33+12\sqrt{3}}.$

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