The number of roots of the equation, $\sin x + 2 \sin 2x = 3 + \sin 3x,$ is

We analyze the ranges of the trigonometric expressions by rewriting the equation as sin x - sin 3x + 2 sin 2x = 3. Using the sum-to-product identity, sin x - sin 3x equals -2 cos 2x sin x, and factoring out 2 sin x gives 2 sin x (2 cos x - cos 2x) = 3. The maximum possible value of the expression 2 sin 2x is 2, and the maximum value of (sin x - sin 3x) is 2, meaning their sum can never exceed 4 but realistically cannot reach 3 due to component boundaries, or more simply, using identities the left side simplifies to 4 sin x cos^2 x which has a maximum value of 1.5 at x = pi/4, making it impossible to equal 3. Therefore, there are 0 roots to this equation.

• Thanks
• Wrong explanation