SOLUTION: 8sinx-cosx =4

Question 1074030: 8sinx-cosx =4
Answer by ikleyn(52906) (Show Source): You can put this solution on YOUR website!
.
8sinx - cosx = 4.
~~~~~~~~~~~~~~~~~~~~~~

Introduce new variable y = cos(x).

Then your equation becomes

 = 4 + y.

Square both sides.

64(1-y^2) = 16 + 8y + y^2,

65y^2 + 8y - 48 = 0.

 =  = .


1.   =  =   --->  cos(x) =   --->  x = +/-  + , k = 0, +/-1, +/-2, . . . 


    Check by substituting x into the original equation. 

             Only the solution in the second quadrant works: x = +  + , k = 0, +/-1, +/-2, . . . 

             The solution in the third quadrant, x = -  + , k = 0, +/-1, +/-2, . . . doesn't work (is extraneous).



2.   =  =   --->  cos(x) = {{4/5}}}  --->  x = +/-  + , k = 0, +/-1, +/-2, . . . 


    Check by substituting x into the original equation. 

             Only the solution in the first quadrant works: x = +  + , k = 0, +/-1, +/-2, . . . 

             The solution in the fourth quadrant, x = -  + , k = 0, +/-1, +/-2, . . . doesn't work (is extraneous).


Answer.  The solutions are these angles 

          + , k = 0, +/-1, +/-2, . . .   and    + , k = 0, +/-1, +/-2, . . .

Plots y = 8sin(x) - cos(x) (red) and y = 4 (green)

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