From AutoMATHic

Geometry Level 4

Let the point \((x,y) \) be a point along the unit circle \(x^2+y^2=1\) in the first quadrant and \(\theta\) be the angle measured counterclockwise from the positive \(x\)-axis such that \( \theta = \cos^{-1} \left( \dfrac{4x+3y}5 \right) \).

If \(\tan \theta = \dfrac ab \), where \(a\) and \(b\) are coprime positive integers, find the value of \(a^2+b^2\).

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