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Sagot :
To determine how many possible triangles can be created given [tex]\( \angle A = 8 - \frac{\pi}{6} \)[/tex], [tex]\( c = 10 \)[/tex], and [tex]\( b = 5 \)[/tex], we need to analyze the information using the Law of Sines.
1. Establish the Given Information:
- Angle [tex]\( A \)[/tex] given in radians: [tex]\( A = 8 - \frac{\pi}{6} \)[/tex]
- Side [tex]\( c = 10 \)[/tex]
- Side [tex]\( b = 5 \)[/tex]
2. Law of Sines:
The Law of Sines states:
[tex]\[ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \][/tex]
However, we only have information about angle [tex]\( A \)[/tex], side [tex]\( b \)[/tex], and side [tex]\( c \)[/tex]. To use the Law of Sines to find angle [tex]\( B \)[/tex], we need side [tex]\( a \)[/tex], but since [tex]\( a \)[/tex] is not given, we will use the Law of Sines in the following form to find [tex]\( \sin B \)[/tex]:
[tex]\[ \sin A \cdot b = \sin B \cdot c \][/tex]
3. Calculate [tex]\( \sin A \)[/tex]:
Since [tex]\( A = 8 - \frac{\pi}{6} \)[/tex], we find [tex]\( \sin A \)[/tex].
4. Solve for [tex]\( \sin B \)[/tex]:
By rearranging the equation:
[tex]\[ \sin B = \frac{b \cdot \sin A}{c} \][/tex]
Using our given values:
[tex]\[ \sin B = \frac{5 \cdot \sin (8 - \frac{\pi}{6})}{10} = \frac{1}{2} \sin (8 - \frac{\pi}{6}) \][/tex]
5. Consider the Range of Sine Function:
The sine of an angle must be within the range of -1 to 1. If [tex]\( \sin B \)[/tex] falls within this range, we can have either 1 or 2 possible triangles. Specifically, if [tex]\( \sin B \)[/tex] is between -1 and 1:
- There can be two solutions for [tex]\( B \)[/tex]:
- [tex]\( B_1 = \sin^{-1}(\sin B) \)[/tex]
- [tex]\( B_2 = \pi - B_1 \)[/tex]
- Only if both [tex]\( B_1 \)[/tex] and [tex]\( B_2 \)[/tex] are valid angles (i.e., they sum up to less than [tex]\( \pi \)[/tex] when added to [tex]\( A \)[/tex]), there will be 2 triangles.
6. Determine the Number of Possible Triangles:
Given [tex]\( \sin B \)[/tex] meets the criteria to have possible solutions:
- One triangular solution when [tex]\( B = \sin^{-1}(\sin B) \)[/tex]
- Another possible triangular solution considering the supplementary angle [tex]\( \pi - \sin^{-1}(\sin B) \)[/tex]
Given the provided conditions and solving [tex]\( \sin B \)[/tex], the result indicates that there are indeed two feasible triangles that can be constructed. Thus, the number of possible triangles is:
[tex]\[ 2 \][/tex]
1. Establish the Given Information:
- Angle [tex]\( A \)[/tex] given in radians: [tex]\( A = 8 - \frac{\pi}{6} \)[/tex]
- Side [tex]\( c = 10 \)[/tex]
- Side [tex]\( b = 5 \)[/tex]
2. Law of Sines:
The Law of Sines states:
[tex]\[ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \][/tex]
However, we only have information about angle [tex]\( A \)[/tex], side [tex]\( b \)[/tex], and side [tex]\( c \)[/tex]. To use the Law of Sines to find angle [tex]\( B \)[/tex], we need side [tex]\( a \)[/tex], but since [tex]\( a \)[/tex] is not given, we will use the Law of Sines in the following form to find [tex]\( \sin B \)[/tex]:
[tex]\[ \sin A \cdot b = \sin B \cdot c \][/tex]
3. Calculate [tex]\( \sin A \)[/tex]:
Since [tex]\( A = 8 - \frac{\pi}{6} \)[/tex], we find [tex]\( \sin A \)[/tex].
4. Solve for [tex]\( \sin B \)[/tex]:
By rearranging the equation:
[tex]\[ \sin B = \frac{b \cdot \sin A}{c} \][/tex]
Using our given values:
[tex]\[ \sin B = \frac{5 \cdot \sin (8 - \frac{\pi}{6})}{10} = \frac{1}{2} \sin (8 - \frac{\pi}{6}) \][/tex]
5. Consider the Range of Sine Function:
The sine of an angle must be within the range of -1 to 1. If [tex]\( \sin B \)[/tex] falls within this range, we can have either 1 or 2 possible triangles. Specifically, if [tex]\( \sin B \)[/tex] is between -1 and 1:
- There can be two solutions for [tex]\( B \)[/tex]:
- [tex]\( B_1 = \sin^{-1}(\sin B) \)[/tex]
- [tex]\( B_2 = \pi - B_1 \)[/tex]
- Only if both [tex]\( B_1 \)[/tex] and [tex]\( B_2 \)[/tex] are valid angles (i.e., they sum up to less than [tex]\( \pi \)[/tex] when added to [tex]\( A \)[/tex]), there will be 2 triangles.
6. Determine the Number of Possible Triangles:
Given [tex]\( \sin B \)[/tex] meets the criteria to have possible solutions:
- One triangular solution when [tex]\( B = \sin^{-1}(\sin B) \)[/tex]
- Another possible triangular solution considering the supplementary angle [tex]\( \pi - \sin^{-1}(\sin B) \)[/tex]
Given the provided conditions and solving [tex]\( \sin B \)[/tex], the result indicates that there are indeed two feasible triangles that can be constructed. Thus, the number of possible triangles is:
[tex]\[ 2 \][/tex]
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