To solve the equation sin(x)cos(x) = 0 in the interval from 0 to 2π, we can use the property of the product of two functions being equal to zero. This means that at least one of the factors must be equal to zero.
The equation can be rewritten using the identity for sin(2x):
sin(x)cos(x) = 0 can be expressed as 1/2 * sin(2x) = 0.
Now, we want to find the values of x where sin(2x) = 0.
The sine function is equal to zero at integer multiples of π:
2x = nπ where n is an integer.
Now, dividing both sides by 2 gives us:
x = nπ/2.
Next, we need to find appropriate values of n such that x stays within the interval [0, 2π].
Evaluating the equation for consecutive integer values of n:
- For n = 0: x = 0
- For n = 1: x = π/2
- For n = 2: x = π
- For n = 3: x = 3π/2
- For n = 4: x = 2π
Thus, the solutions in the interval 0 ≤ x ≤ 2π are:
- x = 0
- x = π/2
- x = π
- x = 3π/2
- x = 2π
These are all the points where the sine and cosine functions yield zero, confirming that we’ve found all the solutions for the given equation in the specified interval.