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Question Number 48526 by Pk1167156@gmail.com last updated on 25/Nov/18
The maximum and minimum values  of   a cos 2θ+ b sin 2θ  are
$$\mathrm{The}\:\mathrm{maximum}\:\mathrm{and}\:\mathrm{minimum}\:\mathrm{values} \\ $$$$\mathrm{of}\:\:\:{a}\:\mathrm{cos}\:\mathrm{2}\theta+\:{b}\:\mathrm{sin}\:\mathrm{2}\theta\:\:\mathrm{are} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 25/Nov/18
a=rsinα   b=rcosα  a^2 +b^2 =r^2   so r=(√(a^2 +b^2 ))   S=acos2θ+bsin2θ    =rsinαcos2θ+rcosαsin2θ  =rsin(α+2θ)  max value of sin(α+2θ)=1  min value of sin(α+2θ)=−1  so max value of acos2θ+bsin2θ  is   =r×1  =(√(a^2 +b^2 ))    min value of acos2θ+bsin2θ  =r×−1  =−(√(a^2 +b^2 ))
$${a}={rsin}\alpha\:\:\:{b}={rcos}\alpha \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} ={r}^{\mathrm{2}} \:\:{so}\:{r}=\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\: \\ $$$${S}={acos}\mathrm{2}\theta+{bsin}\mathrm{2}\theta \\ $$$$\:\:={rsin}\alpha{cos}\mathrm{2}\theta+{rcos}\alpha{sin}\mathrm{2}\theta \\ $$$$={rsin}\left(\alpha+\mathrm{2}\theta\right) \\ $$$${max}\:{value}\:{of}\:{sin}\left(\alpha+\mathrm{2}\theta\right)=\mathrm{1} \\ $$$${min}\:{value}\:{of}\:{sin}\left(\alpha+\mathrm{2}\theta\right)=−\mathrm{1} \\ $$$${so}\:{max}\:{value}\:{of}\:{acos}\mathrm{2}\theta+{bsin}\mathrm{2}\theta\:\:{is}\: \\ $$$$={r}×\mathrm{1} \\ $$$$=\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\:\: \\ $$$${min}\:{value}\:{of}\:{acos}\mathrm{2}\theta+{bsin}\mathrm{2}\theta \\ $$$$={r}×−\mathrm{1} \\ $$$$=−\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\: \\ $$$$ \\ $$
Answered by ajfour last updated on 25/Nov/18
 acos 2θ+bsin 2θ               = (√(a^2 +b^2 ))sin (tan^(−1) (a/b)+2θ)  so maximum value = (√(a^2 +b^2 ))  and minimum value = −(√(a^2 +b^2 )) .
$$\:{a}\mathrm{cos}\:\mathrm{2}\theta+{b}\mathrm{sin}\:\mathrm{2}\theta\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\:\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\mathrm{sin}\:\left(\mathrm{tan}^{−\mathrm{1}} \frac{{a}}{{b}}+\mathrm{2}\theta\right) \\ $$$${so}\:{maximum}\:{value}\:=\:\sqrt{\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}^{\mathrm{2}} } \\ $$$${and}\:{minimum}\:{value}\:=\:−\sqrt{\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}^{\mathrm{2}} }\:. \\ $$

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