To solve the equation sin(2x) sin(x) = 0, we can set each factor equal to zero.
1. First, we consider the equation sin(2x) = 0. The sine function is zero at integer multiples of π, so we set:
2x = nπ, where n is any integer.
This gives us:
x = nπ/2
2. Next, we look at the equation sin(x) = 0. Similarly, sine is zero at integer multiples of π, so we have:
x = mπ, where m is any integer.
3. Combining these results, the general solutions to the equation sin(2x) sin(x) = 0 are:
- x = nπ/2, where n is any integer.
- x = mπ, where m is any integer.
These solutions can also be expressed in one comprehensive form since both sets of solutions form a subset of values derived from integers. Thus, all x can be expressed as:
x = kπ/2, where k is any integer.
In summary, the complete set of solutions includes all integer multiples of π/2, capturing both sin(x) = 0 and sin(2x) = 0.