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Sagot :
Certainly! Let's solve the equation [tex]\(\tan \theta + \frac{1}{\tan \theta} = 2\)[/tex] within the range [tex]\(0 \leqslant \theta \leqslant 2\pi\)[/tex].
### Step-by-Step Solution:
1. Rewrite the Equation:
Start by rewriting the equation in a simpler form:
[tex]\[ \tan \theta + \cot \theta = 2 \][/tex]
Recall that [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex].
2. Substitution for Simplification:
Let [tex]\(x = \tan \theta\)[/tex]. Then, the equation becomes:
[tex]\[ x + \frac{1}{x} = 2 \][/tex]
3. Combine into a Single Fraction:
Combine the terms on the left side:
[tex]\[ \frac{x^2 + 1}{x} = 2 \][/tex]
4. Eliminate the Fraction:
Multiply both sides by [tex]\(x\)[/tex] to eliminate the fraction:
[tex]\[ x^2 + 1 = 2x \][/tex]
5. Form a Quadratic Equation:
Bring all terms to one side to form a standard quadratic equation:
[tex]\[ x^2 - 2x + 1 = 0 \][/tex]
6. Solve the Quadratic Equation:
Factor the quadratic expression:
[tex]\[ (x - 1)^2 = 0 \][/tex]
Therefore, the solution to the quadratic equation is:
[tex]\[ x = 1 \][/tex]
7. Back-Substitute [tex]\( x \)[/tex]:
Recall that we set [tex]\(x = \tan \theta\)[/tex], so now we have:
[tex]\[ \tan \theta = 1 \][/tex]
8. Find the General Solutions:
For [tex]\(\tan \theta = 1\)[/tex], [tex]\(\theta\)[/tex] can be any angle where the tangent is 1. Knowing that [tex]\(\tan \theta = 1\)[/tex] at [tex]\(\theta = \frac{\pi}{4} + k\pi\)[/tex] for any integer [tex]\(k\)[/tex]:
[tex]\[ \theta = \frac{\pi}{4} + k\pi \][/tex]
9. Determine Solutions in the Given Range:
We need the solutions in the interval [tex]\(0 \leq \theta \leq 2\pi\)[/tex]:
- For [tex]\(k = 0\)[/tex]:
[tex]\[ \theta = \frac{\pi}{4} \approx 0.7854 \quad (\text{which is within the range}) \][/tex]
- For [tex]\(k = 1\)[/tex]:
[tex]\[ \theta = \frac{5\pi}{4} \approx 3.9269908 \quad (\text{which is within the range}) \][/tex]
- For [tex]\(k = 2\)[/tex]:
[tex]\[ \theta = \frac{9\pi}{4} \approx 7.068583 \quad (\text{which is outside the range}) \][/tex]
So, the valid solutions within the given range [tex]\(0 \leq \theta \leq 2\pi\)[/tex] are:
[tex]\[ \theta = \frac{\pi}{4} \quad \text{and} \quad \theta = \frac{5\pi}{4} \][/tex]
### Conclusion:
The solutions to the equation [tex]\(\tan \theta + \frac{1}{\tan \theta} = 2\)[/tex] within the interval [tex]\(0 \leq \theta \leq 2\pi\)[/tex] are:
[tex]\[ \theta = \frac{\pi}{4} \quad (\approx 0.7854) \][/tex]
[tex]\[ \theta = \frac{5\pi}{4} \quad (\approx 3.927) \][/tex]
These are the angles [tex]\(\theta\)[/tex] that satisfy the given equation within the specified range.
### Step-by-Step Solution:
1. Rewrite the Equation:
Start by rewriting the equation in a simpler form:
[tex]\[ \tan \theta + \cot \theta = 2 \][/tex]
Recall that [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex].
2. Substitution for Simplification:
Let [tex]\(x = \tan \theta\)[/tex]. Then, the equation becomes:
[tex]\[ x + \frac{1}{x} = 2 \][/tex]
3. Combine into a Single Fraction:
Combine the terms on the left side:
[tex]\[ \frac{x^2 + 1}{x} = 2 \][/tex]
4. Eliminate the Fraction:
Multiply both sides by [tex]\(x\)[/tex] to eliminate the fraction:
[tex]\[ x^2 + 1 = 2x \][/tex]
5. Form a Quadratic Equation:
Bring all terms to one side to form a standard quadratic equation:
[tex]\[ x^2 - 2x + 1 = 0 \][/tex]
6. Solve the Quadratic Equation:
Factor the quadratic expression:
[tex]\[ (x - 1)^2 = 0 \][/tex]
Therefore, the solution to the quadratic equation is:
[tex]\[ x = 1 \][/tex]
7. Back-Substitute [tex]\( x \)[/tex]:
Recall that we set [tex]\(x = \tan \theta\)[/tex], so now we have:
[tex]\[ \tan \theta = 1 \][/tex]
8. Find the General Solutions:
For [tex]\(\tan \theta = 1\)[/tex], [tex]\(\theta\)[/tex] can be any angle where the tangent is 1. Knowing that [tex]\(\tan \theta = 1\)[/tex] at [tex]\(\theta = \frac{\pi}{4} + k\pi\)[/tex] for any integer [tex]\(k\)[/tex]:
[tex]\[ \theta = \frac{\pi}{4} + k\pi \][/tex]
9. Determine Solutions in the Given Range:
We need the solutions in the interval [tex]\(0 \leq \theta \leq 2\pi\)[/tex]:
- For [tex]\(k = 0\)[/tex]:
[tex]\[ \theta = \frac{\pi}{4} \approx 0.7854 \quad (\text{which is within the range}) \][/tex]
- For [tex]\(k = 1\)[/tex]:
[tex]\[ \theta = \frac{5\pi}{4} \approx 3.9269908 \quad (\text{which is within the range}) \][/tex]
- For [tex]\(k = 2\)[/tex]:
[tex]\[ \theta = \frac{9\pi}{4} \approx 7.068583 \quad (\text{which is outside the range}) \][/tex]
So, the valid solutions within the given range [tex]\(0 \leq \theta \leq 2\pi\)[/tex] are:
[tex]\[ \theta = \frac{\pi}{4} \quad \text{and} \quad \theta = \frac{5\pi}{4} \][/tex]
### Conclusion:
The solutions to the equation [tex]\(\tan \theta + \frac{1}{\tan \theta} = 2\)[/tex] within the interval [tex]\(0 \leq \theta \leq 2\pi\)[/tex] are:
[tex]\[ \theta = \frac{\pi}{4} \quad (\approx 0.7854) \][/tex]
[tex]\[ \theta = \frac{5\pi}{4} \quad (\approx 3.927) \][/tex]
These are the angles [tex]\(\theta\)[/tex] that satisfy the given equation within the specified range.
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