Sin(3x+п/3)=sin(x-П/6)

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To solve this trigonometric equation, we can use the angle addition identity for sine:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Given the equation:
sin(3x + π/3) = sin(x - π/6)

Applying the angle addition identity to both sides, we get:

sin(3x)cos(π/3) + cos(3x)sin(π/3) = sin(x)cos(π/6) - cos(x)sin(π/6)

Now we simplify using the values of cos(π/3) = 1/2, sin(π/3) = √3/2, cos(π/6) = √3/2, sin(π/6) = 1/2:

(√3/2)sin(3x) + (1/2)cos(3x) = (√3/2)sin(x) - (1/2)cos(x)

Multiplying through by 2 to clear the fractions:

√3sin(3x) + cos(3x) = √3sin(x) - cos(x)

Rearranging terms:

√3sin(3x) - √3sin(x) = -cos(3x) + cos(x)

√3sin(3x) - √3sin(x) = -(cos(3x) - cos(x))

Now we can use the sum-to-product identities for sine and cosine:

cos(a) - cos(b) = -2sin((a+b)/2)sin((a-b)/2)

Plugging this into the equation:

√3sin(3x) - √3sin(x) = -2sin((3x + x)/2)sin((3x - x)/2)

Simplifying:

√3sin(2x) = -2sin(2x)*cos(x)

At this point, we can see that either sin(2x) = 0 or cos(x) = -√3/2. From sin(2x) = 0, we get 2x = nπ where n is an integer. From cos(x) = -√3/2, we get x = 5π/6 + 2nπ or x = 7π/6 + 2nπ where n is an integer.

Therefore, the solutions to the equation sin(3x + π/3) = sin(x - π/6) are x = nπ, x = 5π/6 + 2nπ, and x = 7π/6 + 2nπ where n is an integer.