Inverse trigonometric equation. Evaluate without calculator

Question

Sin(2tan^inverse 12÷15)

Answer 1

Remember that the inverse trigonometry is a function that returns an angle. The general approach is to assign an angle as the return value.

In this case, let θ = tan^-1 (12/15). We want to evaluate sin 2θ.

Since tan θ = 12/15, we have sin θ = 12/√12²+15² and cos θ = 15/√12²+15² .

Therefore, sin 2θ = 2sinθcosθ = (2*12*15)/(12²+15²) = 40/41.

Alternatively, if you know the formula sin 2θ = 2tanθ / (1+tan²θ). Then you can simply plug the value and get sin 2θ = 2(12/15) / (1+(12/15)²) = 40/41.