1.

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..

Answer» Here, `sin(B+2C) = -1/2`
As it is a negative value, `(B+2C)` is greater than `180^@` .`:. B+2C = 180+30=> B+2C = 210^@->(1)`
`sin(C-A) = 1/2=> C-A = 30^@->(2)`
`sin(2A+B) = 1/2 => 2A+B = 180-30=>2A+B = 150^@->(3)`
As, `A,B and C` are in AP.
`:. A+C = 2B->(4)`
From (1),
`2B+4C = 420^@`
`=>A+5C = 420^@->(5)`
Adding (2) and (5),
`C-A+A+5C = 420+30=> 6C = 450=> C = 75^@`
`A = 75-30 = 45^@`
`B = (45+75)/2 = 60^@`


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