Q:

How to resolve this equation $\exp(\frac{-tI}{RC}) = \cos(\omega tI)$

Is this possible to resolve this equation. I m interested in expressing $tI$ in function of $R,$ $C,$ and $w.$ Here is the equation : 
$$\exp \left(\frac{-tI}{RC}\right) = \cos(\omega tI)$$ where $tI$ is between $3T/4$ and $T.$
Thank you very much !  

A:

You wish to express tI in terms of R, C and $\omega$ :
I suggest the follwing
$$exp(\frac{-tI}{RC})= cos(\omega tI)$$
Differenciate both sides with respect to (tI) ; You'll get :
$$-\frac{1}{RC}exp(\frac{-tI}{RC})= -\omega sin(\omega tI)$$
Now replace $exp(\frac{-tI}{RC})$ from previous equation,
$$-\frac{1}{RC} cos(\omega tI)=-\omega sin(\omega tI)$$
$$tan(\omega tI)=\frac{1}{RC\omega}$$
$$tI=\frac{1}{\omega} tan^{-1}(\frac{1}{RC\omega}) $$
Hope this helps....... :-)

