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Trigonometric Equations

Equations with trigonometry in them can look scary, but that's nothing that a quick little substitution can't fix.

Level 4

         

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.

It is the case that \(S = \displaystyle\sum_{n=-1}^{2015} \csc((2^{n+4})^{\circ}) = \sec(k^{\circ})\) for some integer \(0 \lt k \lt 90\). Find \(k\).

Find the sum of all \(\theta\) such that \[\sin ^8 {\theta} + \cos ^8 {\theta} = \frac{17}{32} \quad \text{where } 0 \leq \theta \leq \pi. \]

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