If sin = 3/5 and 0 < = x < = pi/2, find the exact value of tan 2θ.

For starters,

tan (2θ) = sin (2θ) / cos (2θ)

and we can expand the sine and cosine using the double angle formulas,

sin (2θ) = 2 sin (θ) cos (θ)

cos (2θ) = 1 - 2sin^2 (θ)

To find sin (2θ), use the Pythagorean identity to compute cos (θ). With θ between 0 and π/2, we know cos (θ) > 0, so

cos^2 (θ) + sin^2 (θ) = 1

==> cos (θ) = √ (1 - sin^2 (θ)) = 4/5

We already know sin (θ), so we can plug everything in:

sin (2θ) = 2 * 3/5 * 4/5 = 24/25

cos (2θ) = 1 - 2 * (3/5) ^2 = 7/25

==> tan (2θ) = (24/25) / (7/25) = 24/7