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Sagot :
To determine the number of solutions to the equation [tex]\(\tan \left(\frac{\theta}{2}\right) = \sin (\theta)\)[/tex] within the interval [tex]\(0 \leq \theta < 2\pi\)[/tex], we need to consider the properties and behaviors of the tangent and sine functions over this interval.
1. Understand the Equation:
We need to solve:
[tex]\[ \tan \left(\frac{\theta}{2}\right) = \sin (\theta) \][/tex]
2. Examine the Range and Behavior:
- The function [tex]\(\tan(x)\)[/tex] has discontinuities (or undefined points) where [tex]\(x = \frac{(2k+1)\pi}{2}\)[/tex] for any integer [tex]\(k\)[/tex].
- [tex]\(\sin(x)\)[/tex] is defined for all [tex]\(x\)[/tex], and ranges between -1 and 1.
3. Solving for Solutions:
Given the characteristics of tangent and sine:
- [tex]\(\sin(\theta)\)[/tex] reaches its maximum (1) and minimum (-1) values at [tex]\(\theta = \frac{\pi}{2}\)[/tex] and [tex]\(\theta = \frac{3\pi}{2}\)[/tex], respectively.
- [tex]\(\tan \left(\frac{\theta}{2}\right)\)[/tex] will be equal to [tex]\(\sin(\theta)\)[/tex] at specific points where both functions align.
4. Identify Valid Solutions:
- We are considering values where both functions are defined and result in equality.
- For [tex]\(\tan \left(\frac{\theta}{2}\right)\)[/tex] to equal [tex]\(\sin(\theta)\)[/tex], [tex]\(\theta\)[/tex] should be carefully examined to ensure it does not fall at points where [tex]\(\tan\)[/tex] or [tex]\(\sin\)[/tex] are undefined.
5. List the Solutions:
From the details, we find three specific solutions within the interval from 0 to less than [tex]\(2\pi\)[/tex]:
[tex]\[ \theta = 0, \theta = \frac{3\pi}{2}, \theta = \frac{\pi}{2} \][/tex]
6. Exclude Any Undefined Solutions:
Each of these solutions must be checked to ensure that neither tangent nor sine functions are undefined at these points:
- At [tex]\(\theta = 0\)[/tex], [tex]\(\tan(0) = 0\)[/tex] and [tex]\(\sin(0) = 0\)[/tex], hence valid.
- At [tex]\(\theta = \frac{\pi}{2}\)[/tex], [tex]\(\tan \left(\frac{\pi}{4}\right) = 1\)[/tex] and [tex]\(\sin\left(\frac{\pi}{2}\right) = 1\)[/tex], hence valid.
- [tex]\(\theta = \frac{3\pi}{2}\)[/tex] results in both [tex]\(\tan\)[/tex] and [tex]\(\sin\)[/tex] values aligning validly in this interval, confirming it is a valid solution.
So, the number of valid solutions is three.
Thus, the answer is:
[tex]\[ \boxed{3} \][/tex]
1. Understand the Equation:
We need to solve:
[tex]\[ \tan \left(\frac{\theta}{2}\right) = \sin (\theta) \][/tex]
2. Examine the Range and Behavior:
- The function [tex]\(\tan(x)\)[/tex] has discontinuities (or undefined points) where [tex]\(x = \frac{(2k+1)\pi}{2}\)[/tex] for any integer [tex]\(k\)[/tex].
- [tex]\(\sin(x)\)[/tex] is defined for all [tex]\(x\)[/tex], and ranges between -1 and 1.
3. Solving for Solutions:
Given the characteristics of tangent and sine:
- [tex]\(\sin(\theta)\)[/tex] reaches its maximum (1) and minimum (-1) values at [tex]\(\theta = \frac{\pi}{2}\)[/tex] and [tex]\(\theta = \frac{3\pi}{2}\)[/tex], respectively.
- [tex]\(\tan \left(\frac{\theta}{2}\right)\)[/tex] will be equal to [tex]\(\sin(\theta)\)[/tex] at specific points where both functions align.
4. Identify Valid Solutions:
- We are considering values where both functions are defined and result in equality.
- For [tex]\(\tan \left(\frac{\theta}{2}\right)\)[/tex] to equal [tex]\(\sin(\theta)\)[/tex], [tex]\(\theta\)[/tex] should be carefully examined to ensure it does not fall at points where [tex]\(\tan\)[/tex] or [tex]\(\sin\)[/tex] are undefined.
5. List the Solutions:
From the details, we find three specific solutions within the interval from 0 to less than [tex]\(2\pi\)[/tex]:
[tex]\[ \theta = 0, \theta = \frac{3\pi}{2}, \theta = \frac{\pi}{2} \][/tex]
6. Exclude Any Undefined Solutions:
Each of these solutions must be checked to ensure that neither tangent nor sine functions are undefined at these points:
- At [tex]\(\theta = 0\)[/tex], [tex]\(\tan(0) = 0\)[/tex] and [tex]\(\sin(0) = 0\)[/tex], hence valid.
- At [tex]\(\theta = \frac{\pi}{2}\)[/tex], [tex]\(\tan \left(\frac{\pi}{4}\right) = 1\)[/tex] and [tex]\(\sin\left(\frac{\pi}{2}\right) = 1\)[/tex], hence valid.
- [tex]\(\theta = \frac{3\pi}{2}\)[/tex] results in both [tex]\(\tan\)[/tex] and [tex]\(\sin\)[/tex] values aligning validly in this interval, confirming it is a valid solution.
So, the number of valid solutions is three.
Thus, the answer is:
[tex]\[ \boxed{3} \][/tex]
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