Explanation of the Function
The given function is a combination of sine and cosine functions: \(5\sin\theta + 12\cos\theta\). To find the maximum value of this function, we need to rewrite it in the form of \(A\sin(\theta + \phi)\), where \(A\) is the amplitude and \(\phi\) is the phase shift.

Using Trigonometric Identities
We can rewrite the function as:
\(5\sin\theta + 12\cos\theta\)
\(= \sqrt{5^2 + 12^2}(\frac{5}{\sqrt{5^2 + 12^2}}\sin\theta + \frac{12}{\sqrt{5^2 + 12^2}}\cos\theta)\)
\(= 13(\frac{5}{13}\sin\theta + \frac{12}{13}\cos\theta)\)

Identifying the Amplitude and Phase Shift
From the rewritten form, we can see that the amplitude is 13. To find the phase shift, we need to convert the function into the form \(A\sin(\theta + \phi)\).

Calculating the Maximum Value
The maximum value of the function \(A\sin(\theta + \phi)\) is \(|A|\). Therefore, the greatest value of the function \(5\sin\theta + 12\cos\theta\) is \(|13| = 13\).
In conclusion, the greatest value of the given function is 13. This value represents the maximum point the function reaches within its periodic cycle.