Solve sin ( 2 x ) cos ( 4 x ) − cos ( 2 x ) sin ( 4 x ) = − 0.75 for the smallest positive solution.

Answer 1

Let w = 2x

Then the equation becomes:

sin(w)cos(2w) - cos(w)sin(2w) = -0.75

Using cos(2w) = 1 - 2sin²(w) and using sin(2w) = 2sin(w)cos(w) we get:

sin(w)(1 - 2sin²(w)) - cos(w)(2sin(w)cos(w)) = -0.75

sin(w) - 2sin3(w) - 2cos²(w)sin(w) = -0.75

Using cos²(w) = 1 - sin²(w) we get:

sin(w) - 2sin3(w) - 2sin(w)(1 - sin²(w)) = -0.75

sin(w) - 2sin3(w) - 2sin(w) + 2sin³(w) = -0.75

-sin(w) = -0.75

sin(w) = 0.75

Using a calculator, w = 0.848 in radian measure (this would be the smallest positive value of w)

But w = 2x so 2x = 0.848 or x = 0.424 (rounded to 3 decimal places)

Answer 2

Graphing is the easiest way to get the answer, and

Sin(2x)Cos(4x)-Cos(2x)Sin(4x) =

Sin(2x-4x) = Sin(2x)Cos(4x) - Cos(2x)Sin(4x) = -.75

" = -.75

Sin(-2x) = -.75

Sin(2x) = .75

2x = 48.590

x=24.295°