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Question

For x(0,π), then equation sin x+sin 2xsin 3x=3 has 


Your Answer
A
Infinitely many solutions
Your Answer
B
Three solutions
Your Answer
C
One solution
Correct Answer
D

No solution


Solution

The correct option is D

No solution


sin x+2sin 2xsin 3x=3sin x+4sinxcosx3sin x+4sin3x=3sin x(2+4cos x+4sin2x)=3sin x(2+4cos x+44cos2x)=32+4cos x4cos2x=3sinx2(4cos2x2.2cosx+1)+1=3sin x3(2cos x1)2=3sin x
LHS 3 and RHS 3
   Only possible scenario for a solution to exist is when LHS = RHS = 3. However, LHS and RHS cannot be simlultaneously equal to 3 for a particular value of x. Hence no solution.

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