Here $a \neq 0 $. Both $a$ and $\theta $ are real numbers.
I tried to consider its Fourier transform and therefore considered the integral
$$ \int_{-\infty}^\infty e^{-ik x } \cos(a|x| + \theta ) dx . $$
The problem is that the integral does not converge. I then tried to regularize it by introducing the factor $e^{-\lambda |x|}$, where $\lambda $ is a positive number which will be taken to $0$ in the end.
By simple and straightforward calculation, I get
$$ \cos \theta \left ( \frac{\lambda}{\lambda^2 + (k-a)^2} + \frac{\lambda}{\lambda^2 + (k+a)^2} \right )+\sin \theta \left( \frac{k-a}{\lambda^2 + (k-a)^2} + \frac{k+a }{\lambda^2 + (k+a)^2} \right). $$
Letting $\lambda \rightarrow 0^+$, I get
$$ \pi \cos \theta (\delta(k-a) + \delta(k+a))+ \sin\theta \left (\frac{P}{k-a} - \frac{P}{k+a} \right). $$
Here $P$ means the principal value.
Is this right?
It should be related to the theory of distribution, right?