First, we must acknowledge that tan(x) = sin(x)/cos(x). So, if tan(x) = 4/3, that means sin(x) and cos(x) are fractions that can be simplified into 4/3 (although we don't know just what these fractions are yet).
Because we're given that sin(x) is positive, and we know that tan(x) is positive too, we can deduce that x is in the first quadrant where both sin(x) and cos(x) are positive.
To find cos(x), we will use the Pythagorean trigonometric identity: sin²(x) + cos²(x) = 1.
To express sin(x) in terms of tan(x), we can rearrange the above relationship:
sin(x) = tan(x)/ sqrt(1 + tan²(x))
We have the value of tan(x), so we can use that to find sin(x):
sin(x) = (4/3)/ sqrt(1 + (4/3)²)
Solving this will yield:
sin(x) = 4/5
Substituting sin(x) = 4/5 into the Pythagorean trigonometric identity, we can solve for cos(x):
Cos(x) = sqrt[1 - sin²(x)]
Using sin(x) = 4/5, we find:
Cos(x) = sqrt[1 - (4/5)²] = 3/5.
Now, we can apply the double angle formula for sine that states sin(2x) = 2sin(x)cos(x)
Substituting the above sin(x) and cos(x) in, we get:
sin(2x) = 2 * (4/5) * (3/5) = 48/25