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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 …
Ifsin4α+cos4β=4sinαcosβ,0α,βπ2,thensinα+cosβequalto
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:sayp=sinα,q=cosβforagivenq,youcansolveforp:p44qp+q4=0thatmeansp+qconstant
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.
nouniquevalue!0sinα+cosβ1.25099sothequestionshouldhavebeen:findthemaximumvalueofsinα+cosβ.
Commented by liberty last updated on 10/Apr/21
answer is 2
answeris2
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!ifsinα+cosβ=2,thensinα=1,cosβ=1andsin4α+cos4β=24sinαcosβ=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.
youcanalsosee:withα=0,β=π2whichfulfullsin4α+cos4β=4sinαcosβ,butsinα+cosβ=02.anotherexample:α=π/3,β=1.40749whichfulfullsin4α+cos4β=4sinαcosβ,butsinα+cosβ=1.02862.theseexamplesshowalsothatsinα+cosβmayhavedifferentvalues.

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