1. The problem involves understanding and comparing two functions: $$f(t) = \cos(at)$$ and $$f(t) = \sinh(at)$$. 2. The cosine function $$\cos(at)$$ is a periodic function with period $$\frac{2\pi}{a}$$, oscillating between -1 and 1. 3. The hyperbolic sine function $$\sinh(at)$$ is defined as $$\sinh(x) = \frac{e^x - e^{-x}}{2}$$ and is not periodic; it grows exponentially for large positive or negative values of $$t$$. 4. To analyze these functions, note: - $$\cos(at)$$ is bounded and oscillatory. - $$\sinh(at)$$ is unbounded and grows without limit. 5. The graph of $$\cos(at)$$ will show waves oscillating between -1 and 1. 6. The graph of $$\sinh(at)$$ will show an increasing curve for positive $$t$$ and a decreasing curve for negative $$t$$, crossing zero at $$t=0$$. 7. These functions have different shapes and behaviors due to their definitions and properties. Final answer: The function $$f(t) = \cos(at)$$ is periodic and oscillatory, while $$f(t) = \sinh(at)$$ is unbounded and grows exponentially.