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sin-2x-sin-2y-4-9-cos-x-y-1-sin-x-y-0-lt-x-lt-pi-2-0-lt-y-lt-pi-2-find-the-value-of-sin-x-y-a-2-3-b-1-3-c-1-9-d-2-9-e-2-3-




Question Number 105243 by bemath last updated on 27/Jul/20
 { ((sin 2x + sin 2y = (4/9))),((cos (x−y) = 1−sin (x+y))) :}  0 < x < (π/2) ; 0 < y < (π/2)  find the value of sin (x+y)  (a) −(2/3)   (b) −(1/3)   (c) (1/9)  (d) (2/9)    (e) (2/3)
{sin2x+sin2y=49cos(xy)=1sin(x+y)0<x<π2;0<y<π2findthevalueofsin(x+y)(a)23(b)13(c)19(d)29(e)23
Answered by bramlex last updated on 27/Jul/20
⇒ 2sin (x+y)cos (x−y)=(4/9)...(1)  ⇒cos (x−y) = 1−sin(x+y)...(2)  let sin (x+y) = v  ⇒2v.(1−v) = (4/9)  2v^2 −2v + (4/9) = 0  v^2 −v + (2/9) = 0  v = ((1 + (√(1−4.((2/9)))))/2) = ((1+(1/3))/2)  ∴ sin (x+y) = (4/6) = (2/3)  note 0 < x+y < π then sin(x+y)>0
2sin(x+y)cos(xy)=49(1)cos(xy)=1sin(x+y)(2)letsin(x+y)=v2v.(1v)=492v22v+49=0v2v+29=0v=1+14.(29)2=1+132sin(x+y)=46=23note0<x+y<πthensin(x+y)>0
Commented by bemath last updated on 27/Jul/20
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afdolll
Answered by Dwaipayan Shikari last updated on 27/Jul/20
sin2x+sin2y  =2sin(x+y)cos(x−y)  ⇒2sin(x+y)(1−sin(x+y))=(4/9)  ⇒2sin^2 (x+y)−2sin(x+y)+(4/9)=0  ⇒sin(x+y)=((2+(√(4−((32)/9))))/4)=((2+(2/3))/4)=(2/3)
sin2x+sin2y=2sin(x+y)cos(xy)2sin(x+y)(1sin(x+y))=492sin2(x+y)2sin(x+y)+49=0sin(x+y)=2+43294=2+234=23

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