How do you solve the equation sin(3x) * cos(2x) = 0?

To solve the equation sin(3x) * cos(2x) = 0, we can use the fact that the product of two functions is zero if at least one of the functions is zero.

This means we need to set each function to zero:

1. Step 1: Solve sin(3x) = 0
2. We know that the sine function equals zero at integer multiples of π:

3x = nπ

3. Where n is any integer.
4. To isolate x, divide both sides by 3:

x = nπ/3

5. So, from this, we have solutions for x as:

x = 0, π/3, 2π/3, π, ...

6. Step 2: Solve cos(2x) = 0
7. The cosine function equals zero at odd multiples of π/2:

2x = (2m + 1)π/2

8. Where m is any integer.
9. Now, divide both sides by 2 to solve for x:

x = (2m + 1)π/4

10. So, solutions for x from this equation are:

x = π/4, 3π/4, 5π/4, 7π/4, ...

Final Solutions:

Combining both sets of solutions, we have:

x = nπ/3 and x = (2m + 1)π/4

Where n and m are integers. This gives us a comprehensive set of solutions for the original equation sin(3x) * cos(2x) = 0.