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Geometry

Trigonometric Equations

Trigonometric Equations: Level 3 Challenges

         

Find the sum, in degrees, of all \(\theta\) between \(0^{\circ}\) and \(360^{\circ}\) such that

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

If \(\sec x + \tan x = \dfrac{22}{7}\) and \(\csc x + \cot x = \dfrac{m}{n}\), where \(m\) and \(n\) are coprime positive integers, find \(m+n\).

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

The sum of all positive solutions of \[ 2\cos 2x \left( \cos 2x- \cos\frac { 2014 \pi ^ 2 }{ x } \right) =\cos { 4x } -1\] is \(k\pi \). Find \(k\).

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

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