Geometry

Trigonometric Equations

Trigonometric Equations: Level 3 Challenges

         

Find the sum, in degrees, of all θ\theta between 00^{\circ} and 360360^{\circ} such that

tan(2θ330)=3.\tan(2\theta-330^{\circ})=\sqrt{3}.

If secx+tanx=227\sec x + \tan x = \dfrac{22}{7} and cscx+cotx=mn\csc x + \cot x = \dfrac{m}{n}, where mm and nn are coprime positive integers, find m+nm+n.

The smallest positive solution of the equation 2sin23xcos8x1=02 \sin^2 3x - \cos 8x - 1 = 0 in the interval (0,π2)\left( 0, \frac{\pi}{2}\right) can be expressed in the form aπb\frac{a\pi}{b}, where aa and bb are coprime positive integers, find a+ba+b.

The sum of all positive solutions of 2cos2x(cos2xcos2014π2x)=cos4x1 2\cos 2x \left( \cos 2x- \cos\frac { 2014 \pi ^ 2 }{ x } \right) =\cos { 4x } -1 is kπk\pi . Find kk.

cos(psinx)=sin(pcosx)\large \cos(p \cdot \sin x)=\sin(p \cdot \cos x) Find the smallest positive integer pp for which the above equation has a solution for x[0,2π]x \in [0, 2\pi].

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