Revisiting familiar triangles

Geometry Level 5

\[ \large \sqrt{5-4\cos(x)} + \sqrt{13-12\sin(x)} = \sqrt{10} \]

If \(x\) is a positive acute angle that satisfies the equation above, and can be represented in the form of \( \cos^{-1} \left( \dfrac{a-\sqrt b}c \right) \) for positive integers \(a,b\) and \(c\) with \(b\) square-free, find the value of \(a+b+c\).

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