Trigonometric Equations

If  0 \(\le\) x < \({\pi \over 2}\),  then the number of values of x for which sin x \(-\) sin 2x + sin 3x = 0, is :

Let S = {\(\theta\) \(\in\) [–2\(\pi\), 2\(\pi\)] : 2cos2\(\theta\) + 3sin\(\theta\) = 0}.
Then the sum of the elements of S is

The number of solutions of the equation
1 + sin4
x = cos23x, \(x \in \left[ { - {{5\pi } \over 2},{{5\pi } \over 2}} \right]\) is :

Let S be the set of all \(\alpha\) \(\in\) R such that the equation, cos2x + \(\alpha\)sinx = 2\(\alpha\)– 7 has a solution. Then S is equal to :

If [x] denotes the greatest integer \(\le\) x, then the system of linear equations [sin \(\theta\)]x + [–cos\(\theta\)]y = 0, [cot\(\theta\)]x + y = 0

The sum of all values of \(\theta\) \(\in\)\(\left( {0,{\pi \over 2}} \right)\) satisfying sin2 2\(\theta\) + cos4 2\(\theta\) = \({3 \over 4}\) is -

The number of solutions of sin3x = cos 2x, in the interval \(\left( {{\pi \over 2},\pi } \right)\) is :

If sum of all the solutions of the equation
\(8\cos x.\left( {\cos \left( {{\pi \over 6} + x} \right).\cos \left( {{\pi \over 6} - x} \right) - {1 \over 2}} \right) = 1\)
in [0, \(\pi\)] is k\(\pi\), then k is equal to

The number of x \(\in\) [0,   2\(\pi\) ] for which
\(\left| {\sqrt {2{{\sin }^4}x + 18{{\cos }^2}x} - \sqrt {2{{\cos }^4}x + 18{{\sin }^2}x} } \right| = 1\) is :

If \(0 \le x < 2\pi\), then the number of real values of \(x\), which satisfy the equation \(\,\cos x + \cos 2x + \cos 3x + \cos 4x = 0\) is:

The number of values of \(x\) in the interval \(\left[ {0,3\pi } \right]\,\) satisfying the equation \(2{\sin ^2}x + 5\sin x - 3 = 0\) is

The number of solution of \(\tan \,x + \sec \,x = 2\cos \,x\) in \(\left[ {0,\,2\,\pi } \right]\) is