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Question Number 153736 by ZiYangLee last updated on 09/Sep/21

Given that 7 cos 2θ+24 sin^2 θ=R cos(2θ−α), where R>0 and 0<α<(π/2), find the maximum value of 14 cos^2 θ+48 sin θ cos θ.

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{7}\:\mathrm{cos}\:\mathrm{2}\theta+\mathrm{24}\:\mathrm{sin}^{\mathrm{2}} \theta={R}\:\mathrm{cos}\left(\mathrm{2}\theta−\alpha\right), \\ $$ $$\mathrm{where}\:{R}>\mathrm{0}\:\mathrm{and}\:\mathrm{0}<\alpha<\frac{\pi}{\mathrm{2}},\:\mathrm{find}\:\mathrm{the}\:\mathrm{maximum} \\ $$ $$\mathrm{value}\:\mathrm{of}\:\mathrm{14}\:\mathrm{cos}^{\mathrm{2}} \theta+\mathrm{48}\:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta. \\ $$

Answered by mr W last updated on 09/Sep/21

7 cos 2θ+24 sin^2 θ=R cos(2θ−α) 7 cos 2θ−12cos 2θ+12=R cos(2θ−α) −5cos 2θ+12=R cos2θ cos α−R sin 2θ sin α 12=(R cos α+5) cos2θ−R sin α sin 2θ ((12)/( (√((Rsin α)^2 +(R cos α+5)^2 ))))=cos ϕ cos2θ−sin ϕ sin 2θ ((12)/( (√(R^2 +25+10R cos α))))=cos(2θ+ϕ) 2θ=cos^(−1) ((12)/( (√(R^2 +25+10R cos α))))−ϕ cos 2θ=((12(R cos α+5))/( R^2 +25+10R cos α))+((R sin α(√(R^2 −119+10R cos α)))/( R^2 +25+10R cos α)) sin 2θ=(((R cos α+5)(√(R^2 −119+10R cos α)))/( R^2 +25+10R cos α))−((12R sin α)/( R^2 +25+10R cos α)) 14 cos^2 θ+48 sin θ cos θ =7 cos 2θ+24 sin 2θ+7 =((7(12(R cos α+5)+R sin α(√(R^2 −119+10R cos α)))+24((R cos α+5)(√(R^2 −119+10R cos α))−12R sin α))/(R^2 +25+10R cos α))+7 =((84R cos α−288R sin α+420+(24R cos α+7R sin α+120)(√(R^2 −119+10R cos α)))/(R^2 +25+10R cos α))+7 .....

$$\mathrm{7}\:\mathrm{cos}\:\mathrm{2}\theta+\mathrm{24}\:\mathrm{sin}^{\mathrm{2}} \theta={R}\:\mathrm{cos}\left(\mathrm{2}\theta−\alpha\right) \\ $$ $$\mathrm{7}\:\mathrm{cos}\:\mathrm{2}\theta−\mathrm{12cos}\:\mathrm{2}\theta+\mathrm{12}={R}\:\mathrm{cos}\left(\mathrm{2}\theta−\alpha\right) \\ $$ $$−\mathrm{5cos}\:\mathrm{2}\theta+\mathrm{12}={R}\:\mathrm{cos2}\theta\:\mathrm{cos}\:\alpha−{R}\:\mathrm{sin}\:\mathrm{2}\theta\:\mathrm{sin}\:\alpha \\ $$ $$\mathrm{12}=\left({R}\:\mathrm{cos}\:\alpha+\mathrm{5}\right)\:\mathrm{cos2}\theta−{R}\:\mathrm{sin}\:\alpha\:\mathrm{sin}\:\mathrm{2}\theta \\ $$ $$\frac{\mathrm{12}}{\:\sqrt{\left({R}\mathrm{sin}\:\alpha\right)^{\mathrm{2}} +\left({R}\:\mathrm{cos}\:\alpha+\mathrm{5}\right)^{\mathrm{2}} }}=\mathrm{cos}\:\varphi\:\mathrm{cos2}\theta−\mathrm{sin}\:\varphi\:\mathrm{sin}\:\mathrm{2}\theta \\ $$ $$\frac{\mathrm{12}}{\:\sqrt{{R}^{\mathrm{2}} +\mathrm{25}+\mathrm{10}{R}\:\mathrm{cos}\:\alpha}}=\mathrm{cos}\left(\mathrm{2}\theta+\varphi\right) \\ $$ $$\mathrm{2}\theta=\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{12}}{\:\sqrt{{R}^{\mathrm{2}} +\mathrm{25}+\mathrm{10}{R}\:\mathrm{cos}\:\alpha}}−\varphi \\ $$ $$\mathrm{cos}\:\mathrm{2}\theta=\frac{\mathrm{12}\left({R}\:\mathrm{cos}\:\alpha+\mathrm{5}\right)}{\:{R}^{\mathrm{2}} +\mathrm{25}+\mathrm{10}{R}\:\mathrm{cos}\:\alpha}+\frac{{R}\:\mathrm{sin}\:\alpha\sqrt{{R}^{\mathrm{2}} −\mathrm{119}+\mathrm{10}{R}\:\mathrm{cos}\:\alpha}}{\:{R}^{\mathrm{2}} +\mathrm{25}+\mathrm{10}{R}\:\mathrm{cos}\:\alpha} \\ $$ $$\mathrm{sin}\:\mathrm{2}\theta=\frac{\left({R}\:\mathrm{cos}\:\alpha+\mathrm{5}\right)\sqrt{{R}^{\mathrm{2}} −\mathrm{119}+\mathrm{10}{R}\:\mathrm{cos}\:\alpha}}{\:{R}^{\mathrm{2}} +\mathrm{25}+\mathrm{10}{R}\:\mathrm{cos}\:\alpha}−\frac{\mathrm{12}{R}\:\mathrm{sin}\:\alpha}{\:{R}^{\mathrm{2}} +\mathrm{25}+\mathrm{10}{R}\:\mathrm{cos}\:\alpha} \\ $$ $$ \\ $$ $$\mathrm{14}\:\mathrm{cos}^{\mathrm{2}} \theta+\mathrm{48}\:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta \\ $$ $$=\mathrm{7}\:\mathrm{cos}\:\mathrm{2}\theta+\mathrm{24}\:\mathrm{sin}\:\mathrm{2}\theta+\mathrm{7} \\ $$ $$=\frac{\mathrm{7}\left(\mathrm{12}\left({R}\:\mathrm{cos}\:\alpha+\mathrm{5}\right)+{R}\:\mathrm{sin}\:\alpha\sqrt{{R}^{\mathrm{2}} −\mathrm{119}+\mathrm{10}{R}\:\mathrm{cos}\:\alpha}\right)+\mathrm{24}\left(\left({R}\:\mathrm{cos}\:\alpha+\mathrm{5}\right)\sqrt{{R}^{\mathrm{2}} −\mathrm{119}+\mathrm{10}{R}\:\mathrm{cos}\:\alpha}−\mathrm{12}{R}\:\mathrm{sin}\:\alpha\right)}{{R}^{\mathrm{2}} +\mathrm{25}+\mathrm{10}{R}\:\mathrm{cos}\:\alpha}+\mathrm{7} \\ $$ $$=\frac{\mathrm{84}{R}\:\mathrm{cos}\:\alpha−\mathrm{288}{R}\:\mathrm{sin}\:\alpha+\mathrm{420}+\left(\mathrm{24}{R}\:\mathrm{cos}\:\alpha+\mathrm{7}{R}\:\mathrm{sin}\:\alpha+\mathrm{120}\right)\sqrt{{R}^{\mathrm{2}} −\mathrm{119}+\mathrm{10}{R}\:\mathrm{cos}\:\alpha}}{{R}^{\mathrm{2}} +\mathrm{25}+\mathrm{10}{R}\:\mathrm{cos}\:\alpha}+\mathrm{7} \\ $$ $$..... \\ $$

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