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Question Number 138099 by liberty last updated on 10/Apr/21
If sin^4 α+cos^4 β = 4sin α cos β , 0≤α,β ≤ (π/2) , then sin α+cos β equal to …
$${If}\:\mathrm{sin}\:^{\mathrm{4}} \alpha+\mathrm{cos}\:^{\mathrm{4}} \beta\:=\:\mathrm{4sin}\:\alpha\:\mathrm{cos}\:\beta\:, \\ $$$$\mathrm{0}\leqslant\alpha,\beta\:\leqslant\:\frac{\pi}{\mathrm{2}}\:,\:{then}\:\mathrm{sin}\:\alpha+\mathrm{cos}\:\beta\: \\ $$$${equal}\:{to}\:\ldots \\ $$
Commented by mr W last updated on 10/Apr/21
reason: say p=sin α, q=cos β for a given q, you can solve for p: p^4 −4qp+q^4 =0 that means p+q≠constant
$${reason}: \\ $$$${say}\:{p}=\mathrm{sin}\:\alpha,\:{q}=\mathrm{cos}\:\beta \\ $$$${for}\:{a}\:{given}\:{q},\:{you}\:{can}\:{solve}\:{for}\:{p}: \\ $$$${p}^{\mathrm{4}} −\mathrm{4}{qp}+{q}^{\mathrm{4}} =\mathrm{0} \\ $$$${that}\:{means}\:{p}+{q}\neq{constant} \\ $$
Commented by mr W last updated on 10/Apr/21
no unique value! 0≤sin α+cos β≤1.25099 so the question should have been: find the maximum value of sin α+cos β.
$${no}\:{unique}\:{value}! \\ $$$$\mathrm{0}\leqslant\mathrm{sin}\:\alpha+\mathrm{cos}\:\beta\leqslant\mathrm{1}.\mathrm{25099} \\ $$$${so}\:{the}\:{question}\:{should}\:{have}\:{been}: \\ $$$${find}\:{the}\:{maximum}\:{value}\:{of} \\ $$$$\mathrm{sin}\:\alpha+\mathrm{cos}\:\beta. \\ $$
Commented by liberty last updated on 10/Apr/21
answer is 2
$${answer}\:{is}\:\mathrm{2} \\ $$
Commented by mr W last updated on 10/Apr/21
wrong! if sin α+cos β=2, then sin α=1, cos β=1 and sin^4 α+cos^4 β=2 ≠ 4 sin α cos β=4
$${wrong}! \\ $$$${if}\:\mathrm{sin}\:\alpha+\mathrm{cos}\:\beta=\mathrm{2},\:{then} \\ $$$$\mathrm{sin}\:\alpha=\mathrm{1},\:\mathrm{cos}\:\beta=\mathrm{1}\:{and} \\ $$$$\mathrm{sin}^{\mathrm{4}} \:\alpha+\mathrm{cos}^{\mathrm{4}} \:\beta=\mathrm{2}\:\neq\:\mathrm{4}\:\mathrm{sin}\:\alpha\:\mathrm{cos}\:\beta=\mathrm{4} \\ $$
Commented by mr W last updated on 10/Apr/21
you can also see: with α=0, β=(π/2) which fulfull sin^4 α+cos^4 β = 4sin α cos β, but sin α+cos β=0 ≠2. an other example: α=π/3, β=1.40749 which fulfull sin^4 α+cos^4 β = 4sin α cos β, but sin α+cos β=1.0286 ≠2. these examples show also that sin α+cos β may have different values.
$${you}\:{can}\:{also}\:{see}: \\ $$$${with}\:\alpha=\mathrm{0},\:\beta=\frac{\pi}{\mathrm{2}}\:{which}\:{fulfull} \\ $$$$\mathrm{sin}\:^{\mathrm{4}} \alpha+\mathrm{cos}\:^{\mathrm{4}} \beta\:=\:\mathrm{4sin}\:\alpha\:\mathrm{cos}\:\beta,\:{but} \\ $$$$\mathrm{sin}\:\alpha+\mathrm{cos}\:\beta=\mathrm{0}\:\neq\mathrm{2}. \\ $$$$ \\ $$$${an}\:{other}\:{example}: \\ $$$$\alpha=\pi/\mathrm{3},\:\beta=\mathrm{1}.\mathrm{40749}\:{which}\:{fulfull} \\ $$$$\mathrm{sin}\:^{\mathrm{4}} \alpha+\mathrm{cos}\:^{\mathrm{4}} \beta\:=\:\mathrm{4sin}\:\alpha\:\mathrm{cos}\:\beta,\:{but} \\ $$$$\mathrm{sin}\:\alpha+\mathrm{cos}\:\beta=\mathrm{1}.\mathrm{0286}\:\neq\mathrm{2}. \\ $$$$ \\ $$$${these}\:{examples}\:{show}\:{also}\:{that}\: \\ $$$$\mathrm{sin}\:\alpha+\mathrm{cos}\:\beta\:{may}\:{have}\:{different}\:{values}. \\ $$

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