Find All Solutions in the Interval [0, 2π]: cos(x)sin(x)

To find all solutions to the equation cos(x)sin(x) = 0 in the interval [0, 2π], we can break the equation down into simpler parts. The product of cos(x) and sin(x) equals zero if either cos(x) = 0 or sin(x) = 0.

1. **Finding when cos(x) = 0**:
The cosine function equals zero at specific points in the given interval. These occur at:

• x = π/2
• x = 3π/2

2. **Finding when sin(x) = 0**:
The sine function equals zero at:

• x = 0
• x = π
• x = 2π

Now, compiling both sets of solutions from cos(x) and sin(x), we get:

• x = 0
• x = π/2
• x = π
• x = 3π/2
• x = 2π

Thus, the complete set of solutions for cos(x)sin(x) = 0 in the interval [0, 2π] is:

{0, π/2, π, 3π/2, 2π}