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Question Number 106958 by bemath last updated on 08/Aug/20

   @bemath@  (1) Given  { ((x=sin α+sin β)),((y=cos α+cos β)) :}  maximum value of x^2 +y^2  when α=...  (2) find solution set the equation  sin^4 x + sin^4 (x+(π/4))=(1/4) where x ∈ [0,2π]

@bemath@(1)Given{x=sinα+sinβy=cosα+cosβmaximumvalueofx2+y2whenα=...(2)findsolutionsettheequationsin4x+sin4(x+π4)=14wherex[0,2π]

Answered by mr W last updated on 08/Aug/20

(1)  x^2 +y^2 =sin^2  α+sin^2  β+2 sin α sin β+  cos^2  α+cos^2  β+2 cos α cos β  =2+2 cos (α−β)  ⇒(x^2 +y^2 )_(max) =2+2=4  ⇒(x^2 +y^2 )_(min) =2−2=0

(1)x2+y2=sin2α+sin2β+2sinαsinβ+cos2α+cos2β+2cosαcosβ=2+2cos(αβ)(x2+y2)max=2+2=4(x2+y2)min=22=0

Commented by bemath last updated on 08/Aug/20

thank you mr W

thankyoumrW

Commented by bemath last updated on 08/Aug/20

(x^2 +y^2 )_(max)  where cos (α−β)=1  →cos (α−β)= cos 0°  →α−β=0° +k.2π ,⇒α=β+k.2π

(x2+y2)maxwherecos(αβ)=1cos(αβ)=cos0°αβ=0°+k.2π,α=β+k.2π

Answered by mr W last updated on 08/Aug/20

(2)  sin^4  x+(1/4)(sin x+cos x)^4 =(1/4)  sin^4  x+(1/4)(1+2sin xcos x)^2 =(1/4)  sin^4  x+(1/4)(1+4 sin xcos x+4 sin^2  x cos^2  x)=(1/4)  sin^4  x+(1/4)+sin xcos x+ sin^2  x cos^2  x=(1/4)  sin^2  x+sin xcos x=0  sin x(sin x+cos x)=0  (√2)sin x sin (x+(π/4))=0  ⇒sin x=0 ⇒x=kπ  ⇒sin (x+(π/4))=0 ⇒x=kπ−(π/4)

(2)sin4x+14(sinx+cosx)4=14sin4x+14(1+2sinxcosx)2=14sin4x+14(1+4sinxcosx+4sin2xcos2x)=14sin4x+14+sinxcosx+sin2xcos2x=14sin2x+sinxcosx=0sinx(sinx+cosx)=02sinxsin(x+π4)=0sinx=0x=kπsin(x+π4)=0x=kππ4

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