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Sagot :
To solve the problem using the Law of Sines, we follow these steps:
### Given Data:
- [tex]\(\gamma = 46^{\circ}\)[/tex]
- [tex]\(c = 9\)[/tex]
- [tex]\(a = 10\)[/tex]
We need to find all possible triangles that satisfy these conditions.
### Step-by-Step Solution:
1. Convert [tex]\(\gamma\)[/tex] to Radians:
First, convert the given angle [tex]\(\gamma = 46^{\circ}\)[/tex] to radians.
[tex]\[ \gamma_{\text{rad}} = \gamma \times \frac{\pi}{180} = 46 \times \frac{\pi}{180} \text{ radians} \][/tex]
2. Find [tex]\(\sin(\gamma)\)[/tex]:
Using the sine of the angle [tex]\(\gamma_{\text{rad}}\)[/tex].
3. Use Law of Sines to find [tex]\(\sin(\alpha)\)[/tex]:
The Law of Sines states:
[tex]\[ \frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)} \][/tex]
Rearranging to solve for [tex]\(\sin(\alpha)\)[/tex]:
[tex]\[ \sin(\alpha) = \frac{a \sin(\gamma)}{c} \][/tex]
4. Calculate [tex]\(\sin(\alpha)\)[/tex]:
[tex]\[ \sin(\alpha) = \frac{10 \sin(46^{\circ})}{9} \][/tex]
Calculate this value.
5. Check Validity of [tex]\(\sin(\alpha)\)[/tex]:
Ensure that the computed value of [tex]\(\sin(\alpha)\)[/tex] is within the range [tex]\([-1, 1]\)[/tex]. If it is not, then no such triangle exists. In this case, it is valid.
6. Determine [tex]\(\alpha_1\)[/tex] and [tex]\(\beta_1\)[/tex] for the acute triangle:
[tex]\[ \alpha_1 = \sin^{-1}(\sin(\alpha)) \][/tex]
Subtract [tex]\(\alpha_1\)[/tex] and [tex]\(\gamma\)[/tex] from [tex]\(180^{\circ}\)[/tex] to find [tex]\(\beta_1\)[/tex]:
[tex]\[ \beta_1 = 180^{\circ} - \alpha_1 - \gamma \][/tex]
7. Find the length of [tex]\(b\)[/tex] for the acute triangle:
Again using the Law of Sines:
[tex]\[ b_1 = \frac{c \sin(\beta_1)}{\sin(\gamma)} \][/tex]
8. Determine if there is an obtuse triangle:
If [tex]\( \alpha_1 < 90^{\circ}\)[/tex], there is an obtuse triangle, and:
[tex]\[ \alpha_2 = 180^{\circ} - \alpha_1 \][/tex]
Calculate [tex]\(\beta_2\)[/tex]:
[tex]\[ \beta_2 = 180^{\circ} - \alpha_2 - \gamma \][/tex]
Again using the Law of Sines for [tex]\( b_2\)[/tex]:
[tex]\[ b_2 = \frac{c \sin(\beta_2)}{\sin(\gamma)} \][/tex]
### Results:
Using the above steps, we obtain the results:
For the acute triangle:
[tex]\[ \begin{array}{l} \alpha_1 = 53.06^{\circ} \\ \beta_1 = 80.94^{\circ} \\ b_1 = 12.36 \\ \end{array} \][/tex]
For the obtuse triangle:
[tex]\[ \begin{array}{l} \alpha_2 = 126.94^{\circ} \\ \beta_2 = 7.06^{\circ} \\ b_2 = 1.54 \\ \end{array} \][/tex]
Thus, we have the final results:
For the acute angle:
[tex]\[ \begin{array}{l} \alpha_1 = 53.06 \text{ degrees} \\ \beta_1 = 80.94 \text{ degrees} \\ b_1 = 12.36 \end{array} \][/tex]
For the obtuse angle:
[tex]\[ \begin{array}{l} \alpha_2 = 126.94 \text{ degrees} \\ \beta_2 = 7.06 \text{ degrees} \\ b_2 = 1.54 \end{array} \][/tex]
These results represent the possible triangles formed based on the given data.
### Given Data:
- [tex]\(\gamma = 46^{\circ}\)[/tex]
- [tex]\(c = 9\)[/tex]
- [tex]\(a = 10\)[/tex]
We need to find all possible triangles that satisfy these conditions.
### Step-by-Step Solution:
1. Convert [tex]\(\gamma\)[/tex] to Radians:
First, convert the given angle [tex]\(\gamma = 46^{\circ}\)[/tex] to radians.
[tex]\[ \gamma_{\text{rad}} = \gamma \times \frac{\pi}{180} = 46 \times \frac{\pi}{180} \text{ radians} \][/tex]
2. Find [tex]\(\sin(\gamma)\)[/tex]:
Using the sine of the angle [tex]\(\gamma_{\text{rad}}\)[/tex].
3. Use Law of Sines to find [tex]\(\sin(\alpha)\)[/tex]:
The Law of Sines states:
[tex]\[ \frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)} \][/tex]
Rearranging to solve for [tex]\(\sin(\alpha)\)[/tex]:
[tex]\[ \sin(\alpha) = \frac{a \sin(\gamma)}{c} \][/tex]
4. Calculate [tex]\(\sin(\alpha)\)[/tex]:
[tex]\[ \sin(\alpha) = \frac{10 \sin(46^{\circ})}{9} \][/tex]
Calculate this value.
5. Check Validity of [tex]\(\sin(\alpha)\)[/tex]:
Ensure that the computed value of [tex]\(\sin(\alpha)\)[/tex] is within the range [tex]\([-1, 1]\)[/tex]. If it is not, then no such triangle exists. In this case, it is valid.
6. Determine [tex]\(\alpha_1\)[/tex] and [tex]\(\beta_1\)[/tex] for the acute triangle:
[tex]\[ \alpha_1 = \sin^{-1}(\sin(\alpha)) \][/tex]
Subtract [tex]\(\alpha_1\)[/tex] and [tex]\(\gamma\)[/tex] from [tex]\(180^{\circ}\)[/tex] to find [tex]\(\beta_1\)[/tex]:
[tex]\[ \beta_1 = 180^{\circ} - \alpha_1 - \gamma \][/tex]
7. Find the length of [tex]\(b\)[/tex] for the acute triangle:
Again using the Law of Sines:
[tex]\[ b_1 = \frac{c \sin(\beta_1)}{\sin(\gamma)} \][/tex]
8. Determine if there is an obtuse triangle:
If [tex]\( \alpha_1 < 90^{\circ}\)[/tex], there is an obtuse triangle, and:
[tex]\[ \alpha_2 = 180^{\circ} - \alpha_1 \][/tex]
Calculate [tex]\(\beta_2\)[/tex]:
[tex]\[ \beta_2 = 180^{\circ} - \alpha_2 - \gamma \][/tex]
Again using the Law of Sines for [tex]\( b_2\)[/tex]:
[tex]\[ b_2 = \frac{c \sin(\beta_2)}{\sin(\gamma)} \][/tex]
### Results:
Using the above steps, we obtain the results:
For the acute triangle:
[tex]\[ \begin{array}{l} \alpha_1 = 53.06^{\circ} \\ \beta_1 = 80.94^{\circ} \\ b_1 = 12.36 \\ \end{array} \][/tex]
For the obtuse triangle:
[tex]\[ \begin{array}{l} \alpha_2 = 126.94^{\circ} \\ \beta_2 = 7.06^{\circ} \\ b_2 = 1.54 \\ \end{array} \][/tex]
Thus, we have the final results:
For the acute angle:
[tex]\[ \begin{array}{l} \alpha_1 = 53.06 \text{ degrees} \\ \beta_1 = 80.94 \text{ degrees} \\ b_1 = 12.36 \end{array} \][/tex]
For the obtuse angle:
[tex]\[ \begin{array}{l} \alpha_2 = 126.94 \text{ degrees} \\ \beta_2 = 7.06 \text{ degrees} \\ b_2 = 1.54 \end{array} \][/tex]
These results represent the possible triangles formed based on the given data.
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