simplify

sin(2 pi/3+x)

To simplify the expression sin(2π/3 + x), we will use the trigonometric identity for the sum of angles:

sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

In this case, we have A = 2π/3 and B = x. Substituting these values into the formula, we get:

sin(2π/3 + x) = sin(2π/3)cos(x) + cos(2π/3)sin(x)

Now, we need to determine the values of sin(2π/3) and cos(2π/3). To find these values, we can think of a unit circle. At 2π/3 on the unit circle, the x-coordinate is -1/2, and the y-coordinate is √3/2.

So, sin(2π/3) = √3/2 and cos(2π/3) = -1/2.

Substituting these values into the equation, we have:

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

Now, this is the simplified form of sin(2π/3 + x), and it cannot be further simplified without knowing the exact value of x.