SOLUTION: cos(5x)cos(3x)-sin(5x)sin(3x)=sqrt(3)/2, giving the exact solutions which lie in [0,2pi)

Question 1185683: cos(5x)cos(3x)-sin(5x)sin(3x)=sqrt(3)/2, giving the exact solutions which lie in [0,2pi)
Answer by ikleyn(52814) (Show Source): You can put this solution on YOUR website!
.


According to the addition formula for cosine arguments, the left side is  cos(5x+3x) = cos(8x).


Therefore, the given equation is EQUIVALENT to


    cos(8x) = 


and has the solutions for 8x


    8x = ,  ,  ,  , . . . ,   (8 values)

or

    8x = ,  ,  ,  , . . . ,   (another 8 values).


Dividing by 8, we get these 8 different solutions for x


    x = ,  ,  ,  , . . . ,   (8 values)

or

    x = ,  ,  ,  , . . . ,   (another 8 values).


The listed 8 + 8 = 16 values for x represent the full set of solutions to given equation in the interval [0,2pi).


You can make obvios reducing of fractions. I left the fraction in this form in order for you can better see the structure of the solution.

Solved, explained and completed.

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