a simple Inequality with an integral

Question

Reading a thesis it appear the following inequality: $$\frac{N}{2\pi}\int_{-\theta}^{\theta}e^{t\epsilon \cos(s)}ds \leq \frac{\theta N e}{\pi}, $$ where $t \in [\epsilon, \infty)$, $\theta \in (\pi/2,\pi)$. I thought about to use that $\cos(s) \leq 1$, but $$\int_{-\theta}^{\theta}e^{t\epsilon}ds=2\theta e^{t\epsilon}$$ different of $2\theta e$. I suspect that it involves something of polar variables, because the integral begin as a complex integral.