Solve: sin x - 2sin x cos x = 0 for 0 is less than or equal to x is less than 2pi.
To solve the equation sin(x) - 2sin(x)cos(x) = 0, we first factor out sin(x) from the equation:
sin(x)(1 - 2cos(x)) = 0
Now we have two possible cases:
1. sin(x) = 0:
When sin(x) = 0, the possible values for x are multiples of π since sin(0) = sin(π) = sin(2π) = 0. Therefore, x = 0, π, and 2π are solutions.
2. 1 - 2cos(x) = 0:
Solving this equation, we get:
2cos(x) = 1
cos(x) = 1/2
To find the values of x where cos(x) = 1/2, we can look at the unit circle or use the inverse cosine function. The solutions for cos(x) = 1/2 are x = π/3 and x = 5π/3.
Finally, the solutions for the equation sin(x) - 2sin(x)cos(x) = 0 for 0 ≤ x < 2π are:
x = 0, π, 2π, π/3, and 5π/3.