General Solutions of Trigonometric Equations

While determining the number of solutions the equation can have , we use these formulas.

 General solutions to some of the standard equations

Equation	Solution
Sinθ	Θ = nπ , n∈ Z
Cosθ	Θ = (2n+1)π/2 , n∈ Z
Tanθ = 0	Θ = nπ , n∈ Z
Sinθ = 1	Θ = (4n+1)π/2 , n∈ Z
Sinθ = -1	Θ = (4n-1)π/2 , n∈ Z
Cosθ = 1	Θ = 2nπ , n∈ Z
Cosθ = -1	Θ = (2n+1 )π, n∈ Z
Cotθ = 0	Θ = (2n+1 )π/2, n∈ Z

 EQUATIONS OF ANOTHER TYPES

         1.    When the equation is of the form sinθ = sinα
            General solution to the equation is given by θ = n π + (-1)ⁿ , n∈ Z

      2.    When the equation is of the form cosθ = cosα

             General solution is given by θ = 2n π ± α

      3.    When the equation is of the form tanθ = tanα

                  General solution is given by θ = n π + α

SOME IMPORTANT POINTS THAT SHOULD BE REMEMBERED WHILE SOLVING A TRIGONOMETRIC EQUATION 

   1.  While solving a trigonometric equation squaring the equation should be avoided as far as                   possible. If squaring the equation is necessary check the solution for extraneous solution

    2.  Never cancel the terms containing unknown terms which are in a product it may cause loss of             genuine information

     3. Domain of the function should not change if it changes necessary corrections must be made

     4. while solving a trigonometric equation you should check that at any stage the denominator                   doesn’t becomes zero