To solve the equation [tex]\(\sin \left(x - \frac{\pi}{4}\right) = 0\)[/tex] within the interval [tex]\(0 \leq x \leq 2\pi\)[/tex], let's proceed step by step.
### Step 1: Understanding the Sine Function Zero Points
The sine function [tex]\(\sin(\theta) = 0\)[/tex] when [tex]\(\theta = k\pi\)[/tex] for any integer [tex]\(k\)[/tex]. This is because the sine function equals zero at integer multiples of [tex]\(\pi\)[/tex].
### Step 2: Adjusting for the Shift
We need to adjust for the shift in the argument of the sine function. Here our argument is [tex]\(x - \frac{\pi}{4}\)[/tex]. Set it equal to the known zero points of the sine function:
[tex]\[ x - \frac{\pi}{4} = k\pi \][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
Solve for [tex]\(x\)[/tex] by adding [tex]\(\frac{\pi}{4}\)[/tex] to both sides of the equation:
[tex]\[ x = k\pi + \frac{\pi}{4} \][/tex]
### Step 4: Find Solutions Within the Given Range
We need to find values of [tex]\(k\)[/tex] such that [tex]\(0 \leq x \leq 2\pi\)[/tex]. Let's test different integer values of [tex]\(k\)[/tex].
For [tex]\(k=0\)[/tex]:
[tex]\[ x = 0\pi + \frac{\pi}{4} = \frac{\pi}{4} \approx 0.785 \][/tex]
For [tex]\(k=1\)[/tex]:
[tex]\[ x = 1\pi + \frac{\pi}{4} = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \approx 3.927 \][/tex]
For [tex]\(k=-1\)[/tex]:
[tex]\[ x = -1\pi + \frac{\pi}{4} = -\pi + \frac{\pi}{4} = - \frac{3\pi}{4} \][/tex]
[tex]\[- \frac{3\pi}{4}\][/tex] is negative, so it is out of range.
For [tex]\(k=2\)[/tex]:
[tex]\[ x = 2\pi + \frac{\pi}{4} = 2\pi + \frac{\pi}{4} = \frac{9\pi}{4} \][/tex]
[tex]\[\frac{9\pi}{4}\][/tex] is beyond [tex]\(2\pi\)[/tex], so it is out of range.
Checking further values of [tex]\(k\)[/tex] such as [tex]\(k=3, k=-2\)[/tex] will either be out of the range or redundant.
### Step 5: Identify Valid Solutions
Only the solutions for [tex]\(k=0\)[/tex] and [tex]\(k=1\)[/tex] fall within the range [tex]\(0 \leq x \leq 2\pi\)[/tex]:
- For [tex]\(k=0\)[/tex], [tex]\(x = \frac{\pi}{4}\)[/tex].
- For [tex]\(k=1\)[/tex], [tex]\(x = \frac{5\pi}{4}\)[/tex].
### Conclusion
The solutions of the equation [tex]\(\sin \left(x - \frac{\pi}{4}\right) = 0\)[/tex] in the range [tex]\(0 \leq x \leq 2\pi\)[/tex] are:
[tex]\[ x = \frac{\pi}{4} \approx 0.785 \][/tex]
[tex]\[ x = \frac{5\pi}{4} \approx 3.927 \][/tex]
