Geometry
# Trigonometric Equations

Find the smallest possible positive value of \(\theta\) such that \[ \tan^2(\theta)=\sec (\theta).\]

If \(\theta\) can be represented by \(\tan^{-1}\left( \ \sqrt{\dfrac{a+\sqrt{b}}{c}}\ \right)\) where \(b\) is square free and \(a,b,c\) are integers. Find \(a+b+c\).

How many real numbers \(x\) satisfy

\[\pi \cdot \sin x= 2x?\]

A square \(ABCD\) of side length \(k\) contains unit circles at each of corners \(B\) and \(D\) such that each circle is tangent to the square at precisely two points. A ray of light emanating from point \(A\) reflects off each circle and then returns to \(A\), creating a path in the shape of an equilateral triangle.

There is a unique value of \(k\) for which this scenario can occur. Find \(\lfloor 10000\cdot k \rfloor\).

Note: "Reflecting" means that the angle of incidence equals the angle of reflection.

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