Geometry

Trigonometric Equations

Trigonometric Equations: Level 4 Challenges

         

Find the smallest possible positive value of θ\theta such that tan2(θ)=sec(θ). \tan^2(\theta)=\sec (\theta).

If θ\theta can be represented by tan1( a+bc )\tan^{-1}\left( \ \sqrt{\dfrac{a+\sqrt{b}}{c}}\ \right) where bb is square free and a,b,ca,b,c are integers. Find a+b+ca+b+c.

How many real numbers xx satisfy

πsinx=2x?\pi \cdot \sin x= 2x?

A square ABCDABCD of side length kk contains unit circles at each of corners BB and DD such that each circle is tangent to the square at precisely two points. A ray of light emanating from point AA reflects off each circle and then returns to AA, creating a path in the shape of an equilateral triangle.

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

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

It is the case that S=n=12015csc((2n+4))=sec(k)S = \displaystyle\sum_{n=-1}^{2015} \csc((2^{n+4})^{\circ}) = \sec(k^{\circ}) for some integer 0<k<900 \lt k \lt 90. Find kk.

Find the sum of all θ\theta such that sin8θ+cos8θ=1732where 0θπ.\sin ^8 {\theta} + \cos ^8 {\theta} = \frac{17}{32} \quad \text{where } 0 \leq \theta \leq \pi.

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