There are various successive approximations you can use to solve 2t - cos t = 0. I don't know if you'd call them "trial and error".

For instance, Newton's method, where from each guess t you get the next guess t\_new by

t\_new = t - \[ f(t) / f'(t) \]= t - \[ (2t - cos t) /(2 + sin t) \]

If you start with the guess t = 1.0, this converges really rapidly:

t = 1.0

t = 0.486288017

t = 0.450418605

t = 0.450183622

t = 0.450183611