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ABC is a triangle such that sin(2A+B)=s...

ABC is a triangle such that `sin(2A+B)=sin(C-A)=-sin(B+2C)=1/2`. If A,B, and C are in AP. then the value of A,B and C are..

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The correct Answer is:
`A = 45^(@), B = 60^(@), C = 75^(@)`
Given, in `DeltaABC` A, B, C and C are in an AP.
`therefore` A + C = 2B
Also, A + B + C = `180^(@)` implies B = `60^(@)`
and sin (2A + B) = sin (C - A) `=-sin(B+2C)=(1)/(2)" ...(i)"`
`impliessin(2A + 60^(@))=sin(C-A)=-sin(60^(@)+2C)=(1)/(2)`
`implies 2A + 60^(@) = 30^(@), 150^(@)` [neglecting `30^(@)`, as not possible]
`implies 2A + 60^(@) = 150^(@) implies A = 45^(@)`
Again, from Eq. (i),
`sin(60^(@) + 2C) =-1//2`
`implies 60^(@)+2C = 210^(@), 330^(@)`
implies `C = 75^(@) or 135^(@)`
Also, from Eq. (i),
sin (C-A) = `1//2`
`implies C-A = 30^(@), 150^(@)`
For `A = 45^(@), C = 75^(@)`
and `C = 135^(@)` [not possible]
`therefore C = 75^(@)`
Hence, `A = 45^(@), B = 60^(@), C = 75^(@)`
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