I'm trying to estimate the car's motion in a constant velocity scenario with the variable steering angle (i.e, with constant angular velocity). After manipulating the primary equation I got this: $$ \left\{ \begin{array}{cc} e^{k \theta b} & = \cos{(a+bt)} \\ \theta &= \frac{1}{kb}\ln({\cos{(a+bt)}}) \end{array} \right. $$
'θ' is known for a various time instants 't', and 'a','b' are the variables which we need to solve.
Thanks in advance :)
$$e^{k \theta b} = \cos{(a+bt)} $$ No need to write the second equation which is equivalent to the first equation and so gives nothing more.
In your question you said nothing about the parameter $k$. It is known or not ? If it isn't a known constant your problem involves three unknown parameters, not two.
You wrote : 'θ' is known for a various time instants 't'. One can suppose that the number of 'time instants' is more than 2. True ?
Supposing that they are $n$ measurements : $$(t_1,\theta_1), (t_2,\theta_2), ..., (t_k,\theta_k), …, (t_n,\theta_n) $$ the problem is to adjust the parameters $a,b$ (and may be $k$) for the best fit to the experimental points.
If so, the problem is typically a regression problem, more exactly a nonlinear regression problem due to the parameter $b$ appearing in two places in the equation with non-linearity between them and parameter $a$ .
They are a lot of papers dealing with this kind of problem in books and on the web.
Generally solving for approximate of $a$ , $b$ requires iterative method of calculus. For example :
http://mathworld.wolfram.com/NonlinearLeastSquaresFitting.html
https://en.wikipedia.org/wiki/Non-linear_least_squares
Also specialized softwares exist for nonlinear regression and curve fitting. Since publicity for commercial packages is not welcome here, a search on the web with key words "software for nonlinear regression" will give you several proposals.
Anyways, it should be profitable for you in oder to get more pertinent answers to edit your question with complementary information about the difficulty you are facing. $\mathbf{\text{A numerical example with available data should be useful to clarify the problem concretely}}$.