sin3x−sinx=0
To solve the equation sin(3x) - sin(x) = 0, we can use the trigonometric identity that states sin(A) - sin(B) = 2*sin((A-B)/2) * cos((A+B)/2). Applying this identity, we get:
2*sin((3x - x)/2) * cos((3x + x)/2) = 0
Simplifying further:
2*sin(2x/2) * cos(4x/2) = 0
sin(x) * cos(2x) = 0
Now we have two cases to consider:
Case 1: sin(x) = 0
In this case, the solutions will be values of x for which sin(x) = 0. This occurs when x is equal to multiples of π (pi), such as x = 0, π, 2π, etc.
Case 2: cos(2x) = 0
To solve for x when cos(2x) = 0, we need to find the values of x such that cos(2x) = 0. This occurs when 2x = (2n + 1)π/2, where n is an integer.
Solving for x:
2x = (2n + 1)π/2
x = (2n + 1)π/4
So, the solutions for cos(2x) = 0 are x = (2n + 1)π/4 where n is an integer.
In summary, the solutions to the equation sin(3x) - sin(x) = 0 are x = nπ and x = (2n + 1)π/4, where n is an integer.
sin3x = sinx(2cos(2x)+1)
= sinx(2(1-2sin^2x)-1)
= sinx(1-4sin^2x)
= sinx - 4sin^3x
now you have sin^3(x) = 0
or, consider that sin(x)=0 at all multiples of pi.
Sin3x oscillates 3 times as fast, so it is also zero at multiples of pi.