We can rewrite the equation as[8\sin^2(5x) + \frac{1}{2}\sin(10x) + \cos^2(5x) = 4]
Using the double angle formula for sin(2x) and cos(2x)[8\sin^2(5x) + 2\sin(5x)\cos(5x) + \cos^2(5x) = 4]
Since (\sin^2(x) + \cos^2(x) = 1)[8\sin^2(5x) + 2\sin(5x)\cos(5x) + 1 = 4]
Let (u = \sin(5x))[8u^2 + 2u + 1 = 4[8u^2 + 2u - 3 = 0]
Solving this quadratic equation by factoring or using the quadratic formula will give the solutions for (u), which can then be substituted back in for (sin(5x)) to solve for (x).
We can rewrite the equation as
[8\sin^2(5x) + \frac{1}{2}\sin(10x) + \cos^2(5x) = 4]
Using the double angle formula for sin(2x) and cos(2x)
[8\sin^2(5x) + 2\sin(5x)\cos(5x) + \cos^2(5x) = 4]
Since (\sin^2(x) + \cos^2(x) = 1)
[8\sin^2(5x) + 2\sin(5x)\cos(5x) + 1 = 4]
Let (u = \sin(5x))
[8u^2 + 2u + 1 = 4
[8u^2 + 2u - 3 = 0]
Solving this quadratic equation by factoring or using the quadratic formula will give the solutions for (u), which can then be substituted back in for (sin(5x)) to solve for (x).