SOLUTION: Please Find the general solution of: sin3x-sinx= cos2x

Algebra ->  Trigonometry-basics -> SOLUTION: Please Find the general solution of: sin3x-sinx= cos2x      Log On


   

Question 1031139: Please Find the general solution of:
sin3x-sinx= cos2x
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
Please Find the general solution of:
sin(3x) - sin(x)= cos(2x)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Apply the formula  sin%28alpha%29+-+sin%28beta%29 = 2%2Asin%28%28alpha-beta%29%2F2%29%2Acos%28%28alpha%2Bbeta%29%2F2%29  to the left side.

  (Regarding this formula, see the lesson Addition and subtraction of trigonometric functions in this site).

You will get an equivalent equation

2%2Asin%28x%29%2Acos%282x%29 = cos%282x%29.

Simplify it further:

cos%282x%29%2A%282sin%28x%29-1%29 = 0.

Then you have two equations:

(1)  cos(2x) = 0  --->  2x = 2k%2Api+%2B-+pi%2F2  --->  x = k%2Api+%2B-+pi%2F4, k = 0, +/-1, +/-2, . . . ,   

and

(2)  2*sin(x) = 1  --->  sin(x) = 1%2F2  --->  x = pi%2F6+%2B+2k%2Api  and  x = 5pi%2F6+%2B+2k%2Api, k = 0, +/-1, +/-2, . . . .

The union of the sets (1) and (2) is your solution.