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Question

# The minimum value of 3 cos x + 4 sin x + 8 is (a) 5 (b) 9 (c) 7 (d) 3

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Solution

## $3\mathrm{cos}x+4\mathrm{sin}x+8\phantom{\rule{0ex}{0ex}}\mathrm{first}\mathrm{express}3\mathrm{cos}x+4\mathrm{sin}xasa\mathrm{cos}\left(x+A\right)\phantom{\rule{0ex}{0ex}}3\mathrm{cos}x+4\mathrm{sin}x=a\left[\mathrm{cos}x\mathrm{cos}A-\mathrm{sin}x\mathrm{sin}A\right]\phantom{\rule{0ex}{0ex}}\mathrm{i}.\mathrm{e}3\mathrm{cos}x+4\mathrm{sin}x=a\mathrm{cos}x\mathrm{cos}A-a\mathrm{sin}x\mathrm{sin}A\phantom{\rule{0ex}{0ex}}\mathrm{Now},\mathrm{equate}\mathrm{the}\mathrm{coefficients}\mathrm{of}\mathrm{sin}xand\mathrm{cos}x\phantom{\rule{0ex}{0ex}}\mathrm{we}\mathrm{get},\phantom{\rule{0ex}{0ex}}a\mathrm{cos}A=3\phantom{\rule{0ex}{0ex}}\mathrm{and}-a\mathrm{sin}A=4\phantom{\rule{0ex}{0ex}}\mathrm{i}.\mathrm{e}\frac{a\mathrm{sin}A}{a\mathrm{cos}A}=\frac{-4}{3}\phantom{\rule{0ex}{0ex}}\mathrm{i}.\mathrm{e}\mathrm{tan}A=\frac{-4}{3}\phantom{\rule{0ex}{0ex}}\mathrm{i}.\mathrm{e}A={\mathrm{tan}}^{-1}\left(\frac{-4}{3}\right).\phantom{\rule{0ex}{0ex}}\mathrm{also},{\left(a\mathrm{cos}A\right)}^{2}+{\left(a\mathrm{sin}A\right)}^{2}=9+16=25\phantom{\rule{0ex}{0ex}}\mathrm{i}.\mathrm{e}{a}^{2}=25\phantom{\rule{0ex}{0ex}}\mathrm{i}.\mathrm{e}a=5\phantom{\rule{0ex}{0ex}}⇒3\mathrm{cos}x+4\mathrm{sin}x=5\mathrm{cos}\left(x-{\mathrm{tan}}^{-1}\left(\frac{-4}{3}\right)\right)\phantom{\rule{0ex}{0ex}}\mathrm{i}.\mathrm{e}3\mathrm{cos}x+4\mathrm{sin}x+8=5\mathrm{cos}\left(x-{\mathrm{tan}}^{-1}\left(\frac{-4}{3}\right)\right)+8\phantom{\rule{0ex}{0ex}}\mathrm{Since}\mathrm{minimum}\mathrm{value}\mathrm{of}\mathrm{cos}\theta =-1\phantom{\rule{0ex}{0ex}}⇒\left(3\mathrm{cos}x+4\mathrm{sin}x+8\right)\mathrm{min}=5\left(-1\right)+8\phantom{\rule{0ex}{0ex}}\mathrm{i}.\mathrm{e}\mathrm{Minimum}\mathrm{vlaue}\mathrm{of}3\mathrm{cos}x+4\mathrm{sin}x+8=3\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{the}\mathrm{correct}\mathrm{answer}\mathrm{is}\mathrm{option}\mathrm{D}.$

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