To find all solutions to the equation cos x sin 2x = 0 in the interval 0 < x < 2π, we can set each part of the product to zero and solve for x.
1. The first part of the equation is cos x = 0. The cosine function is zero at:
- x = π/2
- x = 3π/2
2. The second part of the equation is sin 2x = 0. The sine function is zero at:
- 2x = nπ, where n is an integer
Dividing by 2 gives:
- x = nπ/2
For the values of n within our interval (0 < x < 2π), we consider:
- n = 1: x = π/2
- n = 2: x = π
- n = 3: x = 3π/2
- n = 4: x = 2π (this is not included as it does not satisfy the open interval)
Putting it all together, the solutions in the interval 0 < x < 2π are:
- x = π/2
- x = π
- x = 3π/2
Thus, the full set of solutions to the equation is: x = π/2, π, 3π/2.