To prove that sin(πx) * sin(x) equals 0 for certain values of x, let’s first analyze the properties of the sine function and the specific conditions under which this product results in zero.
The product of the sine functions sin(πx) and sin(x) will equal zero if either of the sine functions is zero. Therefore, we need to find the values of x that satisfy either of the following equations:
- sin(πx) = 0
- sin(x) = 0
1. **Finding roots of sin(πx):**
The sine function equals zero at integer multiples of π. Thus, we can express sin(πx) as:
sin(πx) = 0 when πx = nπ where n is any integer.
From this, we derive:
x = n, where n is any integer (0, ±1, ±2, …)
2. **Finding roots of sin(x):**
Similarly, sin(x) equals 0 at integer multiples of π, which can be expressed as:
sin(x) = 0 when x = mπ, where m is any integer.
In summary, the product sin(πx) * sin(x) will be zero for:
- x = n (for all integers n)
- x = mπ (for all integers m)
Thus, we conclude that sin(πx) * sin(x) = 0 whenever x is an integer or a multiple of π. In those cases, at least one of the sine functions becomes zero, proving the statement.