1. The problem asks for the argument (angle in the complex plane) of the hyperbolic cosine function, expressed as $\arg(\cosh z)$ for some complex number $z$. 2. Recall the definition of hyperbolic cosine: $$\cosh z = \frac{e^z + e^{-z}}{2}.$$ For real $z$, $\cosh z$ is always real and positive, so $\arg(\cosh z) = 0$. 3. For complex $z = x + iy$, observe that: $$\cosh z = \cosh(x + iy) = \cosh x \cos y + i \sinh x \sin y.$$ 4. The argument is given by: $$\arg(\cosh z) = \arctan \left( \frac{\sinh x \sin y}{\cosh x \cos y} \right).$$ 5. This formula calculates the angle of the complex number $\cosh z$ in the complex plane, relative to the positive real axis. 6. Thus, $\boxed{\arg \cosh z = \arctan \left( \frac{\sinh x \sin y}{\cosh x \cos y} \right)}$. This holds for all $z = x + iy \in \mathbb{C}$.