Question

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

Answer 1

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