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Sagot :
Certainly! Here is a step-by-step solution to the problem:
We need to solve the equation:
[tex]\[ \tan \left(\frac{3\pi}{4} - 2x \right) = -1 \][/tex]
### Step-by-Step Solution
#### Step 1: Understand the behavior of the tangent function
The tangent function [tex]\(\tan(\theta)\)[/tex] equals [tex]\(-1\)[/tex] at angles that are odd multiples of [tex]\(\frac{3\pi}{4}\)[/tex]. Thus:
[tex]\[ \frac{3\pi}{4} - 2x = n\pi + \frac{3\pi}{4} \][/tex]
where [tex]\(n\)[/tex] is any integer.
#### Step 2: Solve for [tex]\(x\)[/tex]
Rearranging the equation:
[tex]\[ 2x = \frac{3\pi}{4} - (n\pi + \frac{3\pi}{4}) \][/tex]
[tex]\[ 2x = -n\pi \][/tex]
[tex]\[ x = -\frac{n\pi}{2} \][/tex]
We need [tex]\(x\)[/tex] to lie within the interval [tex]\([0, \pi]\)[/tex].
[tex]\[ 0 \leq -\frac{n\pi}{2} \leq \pi \][/tex]
[tex]\[ 0 \leq -n \leq 2 \][/tex]
#### Step 3: Determine specific integer values for [tex]\(n\)[/tex]
From the inequality [tex]\(0 \leq -n \leq 2\)[/tex], we find:
[tex]\[ 0 \geq n \geq -2 \][/tex]
So, [tex]\(n\)[/tex] can be [tex]\(-2\)[/tex], [tex]\(-1\)[/tex], or [tex]\(0\)[/tex].
#### Step 4: Calculate the corresponding [tex]\(x\)[/tex] values
1. For [tex]\(n = 0\)[/tex]:
[tex]\[ x = 0 \][/tex]
2. For [tex]\(n = -1\)[/tex]:
[tex]\[ x = -\left(-\frac{\pi}{2}\right) = \frac{\pi}{2} \][/tex]
3. For [tex]\(n = -2\)[/tex]:
[tex]\[ x = -\left(-\frac{2\pi}{2}\right) = \pi \][/tex]
Thus, the solutions to the equation [tex]\(\tan\left(\frac{3\pi}{4} - 2x\right) = -1\)[/tex] within the interval [tex]\([0, \pi]\)[/tex] are [tex]\(0\)[/tex], [tex]\(\frac{\pi}{2}\)[/tex], and [tex]\(\pi\)[/tex].
#### Step 5: Verify the solutions
Each of these values can be verified by substituting back into the original equation to ensure they satisfy [tex]\(\tan\left(\frac{3\pi}{4} - 2x\right) = -1\)[/tex].
### Conclusion
The set of solutions that satisfies the given equation over the interval [tex]\([0, \pi]\)[/tex] is:
[tex]\[ \left\{0, \frac{\pi}{2}, \pi \right\} \][/tex]
### Final Answer
[tex]\[ \boxed{\left\{0, \frac{\pi}{2}, \pi\right\}} \][/tex]
We need to solve the equation:
[tex]\[ \tan \left(\frac{3\pi}{4} - 2x \right) = -1 \][/tex]
### Step-by-Step Solution
#### Step 1: Understand the behavior of the tangent function
The tangent function [tex]\(\tan(\theta)\)[/tex] equals [tex]\(-1\)[/tex] at angles that are odd multiples of [tex]\(\frac{3\pi}{4}\)[/tex]. Thus:
[tex]\[ \frac{3\pi}{4} - 2x = n\pi + \frac{3\pi}{4} \][/tex]
where [tex]\(n\)[/tex] is any integer.
#### Step 2: Solve for [tex]\(x\)[/tex]
Rearranging the equation:
[tex]\[ 2x = \frac{3\pi}{4} - (n\pi + \frac{3\pi}{4}) \][/tex]
[tex]\[ 2x = -n\pi \][/tex]
[tex]\[ x = -\frac{n\pi}{2} \][/tex]
We need [tex]\(x\)[/tex] to lie within the interval [tex]\([0, \pi]\)[/tex].
[tex]\[ 0 \leq -\frac{n\pi}{2} \leq \pi \][/tex]
[tex]\[ 0 \leq -n \leq 2 \][/tex]
#### Step 3: Determine specific integer values for [tex]\(n\)[/tex]
From the inequality [tex]\(0 \leq -n \leq 2\)[/tex], we find:
[tex]\[ 0 \geq n \geq -2 \][/tex]
So, [tex]\(n\)[/tex] can be [tex]\(-2\)[/tex], [tex]\(-1\)[/tex], or [tex]\(0\)[/tex].
#### Step 4: Calculate the corresponding [tex]\(x\)[/tex] values
1. For [tex]\(n = 0\)[/tex]:
[tex]\[ x = 0 \][/tex]
2. For [tex]\(n = -1\)[/tex]:
[tex]\[ x = -\left(-\frac{\pi}{2}\right) = \frac{\pi}{2} \][/tex]
3. For [tex]\(n = -2\)[/tex]:
[tex]\[ x = -\left(-\frac{2\pi}{2}\right) = \pi \][/tex]
Thus, the solutions to the equation [tex]\(\tan\left(\frac{3\pi}{4} - 2x\right) = -1\)[/tex] within the interval [tex]\([0, \pi]\)[/tex] are [tex]\(0\)[/tex], [tex]\(\frac{\pi}{2}\)[/tex], and [tex]\(\pi\)[/tex].
#### Step 5: Verify the solutions
Each of these values can be verified by substituting back into the original equation to ensure they satisfy [tex]\(\tan\left(\frac{3\pi}{4} - 2x\right) = -1\)[/tex].
### Conclusion
The set of solutions that satisfies the given equation over the interval [tex]\([0, \pi]\)[/tex] is:
[tex]\[ \left\{0, \frac{\pi}{2}, \pi \right\} \][/tex]
### Final Answer
[tex]\[ \boxed{\left\{0, \frac{\pi}{2}, \pi\right\}} \][/tex]
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