To solve the equation sinx - cosx = sqrt(2)sin(6x), we can rewrite the right side using the angle addition formula for sine.
sinx - cosx = sqrt(2)(sinxcos(6x) + cosxsin(6x))
Expanding the right side further:
sinx - cosx = sqrt(2)(sinxcos(6x) + cosxsin(6x)= sqrt(2)(sinxcosxcos5x + cosxsinxcos5x= sqrt(2)(sinxcosxcos5x + sinxcosxcos(π/2-5x)= sqrt(2)(sinxcosx(cos5x + cos(π/2-5x))= sqrt(2)(sinxcosx(2cos(π/4)cos(5x-π/4))= sqrt(2)(sinxcosx(cos(5x-π/4))= sqrt(2)sinxcosx*cos(5x-π/4)
Now, we have:
sinx - cosx = sqrt(2)sinxcosxcos(5x-π/4)
Since sinx - cosx = sqrt(2)sinxcosxcos(5x-π/4), we can divide both sides by sinx*cosx to get:
1 = sqrt(2)cos(5x-π/4)
Now, isolating cos(5x-π/4):
cos(5x-π/4) = 1/sqrt(25x - π/4 = ± π/4 + 2πn
Solving for x:
5x = π/2, 3π/2, π/4 + 2πn, 5π/4 + 2πx = π/10, 3π/10, π/20 + 2πn, 5π/20 + 2πn
Therefore, the solutions to the equation sinx - cosx = sqrt(2)sin(6x) are x = π/10, 3π/10, π/20 + 2πn, and 5π/20 + 2πn, where n is an integer.
To solve the equation sinx - cosx = sqrt(2)sin(6x), we can rewrite the right side using the angle addition formula for sine.
sinx - cosx = sqrt(2)(sinxcos(6x) + cosxsin(6x))
Expanding the right side further:
sinx - cosx = sqrt(2)(sinxcos(6x) + cosxsin(6x)
= sqrt(2)(sinxcosxcos5x + cosxsinxcos5x
= sqrt(2)(sinxcosxcos5x + sinxcosxcos(π/2-5x)
= sqrt(2)(sinxcosx(cos5x + cos(π/2-5x))
= sqrt(2)(sinxcosx(2cos(π/4)cos(5x-π/4))
= sqrt(2)(sinxcosx(cos(5x-π/4))
= sqrt(2)sinxcosx*cos(5x-π/4)
Now, we have:
sinx - cosx = sqrt(2)sinxcosxcos(5x-π/4)
Since sinx - cosx = sqrt(2)sinxcosxcos(5x-π/4), we can divide both sides by sinx*cosx to get:
1 = sqrt(2)cos(5x-π/4)
Now, isolating cos(5x-π/4):
cos(5x-π/4) = 1/sqrt(2
5x - π/4 = ± π/4 + 2πn
Solving for x:
5x = π/2, 3π/2, π/4 + 2πn, 5π/4 + 2π
x = π/10, 3π/10, π/20 + 2πn, 5π/20 + 2πn
Therefore, the solutions to the equation sinx - cosx = sqrt(2)sin(6x) are x = π/10, 3π/10, π/20 + 2πn, and 5π/20 + 2πn, where n is an integer.