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Given tan(x)=4/3 and sin(x) is positive, find the sin(2x)

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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

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