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Question

What is the value of y if sin(x+y) = 1/2 sin(x) + √(3)/2 cos(x)?

Answer

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

Using the sum-to-product formula for sine, the given equation is rewritten as sin(x + y) = sin(x) cos(y) + cos(x) sin(y). By comparing this with sin(x+y) = 1/2 sin(x) + √(3)/2 cos(x), we get sin(x) cos(y) - 1/2 sin(x) = - √(3)/2 cos(x) - cos(x) sin(y). Factoring sin(x) out of the terms on the left-hand side and solving for cos(y) yields cos(y) = (-√(3)/2 - sin(y)) / tan(x) + 1/2. Simplifying this expression by using the fact that tan(x) = sin(x) / cos(x) results in cos(y) = (-√(3)/2 - sin(y)) / sin(x) + cos(x)/2. Multiplying both sides by sin(x) and simplifying, we get: cos(y) sin(x) + √(3)/2 cos(x) = -1/2 sin(y) sin(x) + cos(x)/2 sin(x). Using the sum-to-product formula for cosine, we get cos(y) sin(x) +√(3)/2 cos(x) = -1/2 sin(y + x). Solving for y using the inverse sine function, we get y = arcsin(-√(3)/2 cos(x) - 1/2 cos(y) sin(x)) - x