To solve the equation sin x cos x = 0, we can utilize the property that the product of two terms equals zero when at least one of the terms is zero.
This gives us two separate equations to consider:
- sin x = 0
- cos x = 0
Let’s solve each of these cases:
1. Solving sin x = 0
The sine function equals zero at integer multiples of π. This can be expressed as:
x = nπ
where n is any integer (n ∈ ℤ).
2. Solving cos x = 0
The cosine function equals zero at odd multiples of π/2. This can be shown as:
x = (2n + 1)π/2
where n is any integer (n ∈ ℤ).
Final Solution
Combining both sets of solutions, we find that:
- x = nπ for any integer n.
- x = (2n + 1)π/2 for any integer n.
Thus, all solutions to the equation sin x cos x = 0 are:
x = nπ or x = (2n + 1)π/2, where n is any integer.