Answer :
To determine the number of possible triangles based on the given parameters, follow these steps:
Step 1: Convert the given angle ZA to radians (if necessary for calculations).
Given:
- [tex]\( ZA = 48^\circ \)[/tex]
- [tex]\( a = 10 \)[/tex] meters
- [tex]\( b = 12 \)[/tex] meters
Step 2: Use the Law of Sines to find the sine of angle B.
According to the Law of Sines:
[tex]\[ \frac{\sin(B)}{b} = \frac{\sin(A)}{a} \][/tex]
Here, [tex]\( A \)[/tex] corresponds to angle ZA:
[tex]\[ \sin(B) = \frac{b \cdot \sin(ZA)}{a} \][/tex]
First, calculate [tex]\(\sin(ZA)\)[/tex]:
[tex]\[ \sin(48^\circ) \approx 0.7431 \][/tex]
Then:
[tex]\[ \sin(B) = \frac{12 \cdot 0.7431}{10} = 0.8917 \][/tex]
Step 3: Check the value of [tex]\(\sin(B)\)[/tex].
- If [tex]\(\sin(B) > 1\)[/tex] or [tex]\(\sin(B) < -1\)[/tex], no triangle exists.
- If [tex]\(\sin(B) = 1\)[/tex] or [tex]\(\sin(B) = -1\)[/tex], only one right triangle exists.
- If [tex]\(0 < \sin(B) < 1\)[/tex], further calculations are required.
In this case:
[tex]\[ 0 < \sin(B) = 0.8917 < 1 \][/tex]
Hence, [tex]\( \sin(B) \)[/tex] is valid, so further calculations are needed.
Step 4: Calculate angle B.
Using the inverse sine function:
[tex]\[ B = \sin^{-1}(0.8917) \approx 63^\circ \][/tex]
Step 5: Determine if there could be a second possible angle for B.
In a triangle, angle B can have another possible value, [tex]\( B'\)[/tex], where:
[tex]\[ B' = 180^\circ - B \][/tex]
[tex]\[ B' = 180^\circ - 63^\circ = 117^\circ \][/tex]
Step 6: Check the possible values for angle C.
For the first scenario:
[tex]\[ C = 180^\circ - ZA - B \][/tex]
[tex]\[ C = 180^\circ - 48^\circ - 63^\circ = 69^\circ \][/tex]
For the second scenario:
[tex]\[ C' = 180^\circ - ZA - B' \][/tex]
[tex]\[ C' = 180^\circ - 48^\circ - 117^\circ = 15^\circ \][/tex]
Step 7: Determine the number of valid triangles.
Both [tex]\( C \ (\approx 69^\circ) \)[/tex] and [tex]\( C' \ (\approx 15^\circ) \)[/tex] are valid angles as they are both positive and less than 180°.
Conclusion:
Since there are two valid sets of angles that satisfy the triangle conditions, there are two possible triangles in this scenario. Hence:
There are two possible triangles based on the given parameters.
Step 1: Convert the given angle ZA to radians (if necessary for calculations).
Given:
- [tex]\( ZA = 48^\circ \)[/tex]
- [tex]\( a = 10 \)[/tex] meters
- [tex]\( b = 12 \)[/tex] meters
Step 2: Use the Law of Sines to find the sine of angle B.
According to the Law of Sines:
[tex]\[ \frac{\sin(B)}{b} = \frac{\sin(A)}{a} \][/tex]
Here, [tex]\( A \)[/tex] corresponds to angle ZA:
[tex]\[ \sin(B) = \frac{b \cdot \sin(ZA)}{a} \][/tex]
First, calculate [tex]\(\sin(ZA)\)[/tex]:
[tex]\[ \sin(48^\circ) \approx 0.7431 \][/tex]
Then:
[tex]\[ \sin(B) = \frac{12 \cdot 0.7431}{10} = 0.8917 \][/tex]
Step 3: Check the value of [tex]\(\sin(B)\)[/tex].
- If [tex]\(\sin(B) > 1\)[/tex] or [tex]\(\sin(B) < -1\)[/tex], no triangle exists.
- If [tex]\(\sin(B) = 1\)[/tex] or [tex]\(\sin(B) = -1\)[/tex], only one right triangle exists.
- If [tex]\(0 < \sin(B) < 1\)[/tex], further calculations are required.
In this case:
[tex]\[ 0 < \sin(B) = 0.8917 < 1 \][/tex]
Hence, [tex]\( \sin(B) \)[/tex] is valid, so further calculations are needed.
Step 4: Calculate angle B.
Using the inverse sine function:
[tex]\[ B = \sin^{-1}(0.8917) \approx 63^\circ \][/tex]
Step 5: Determine if there could be a second possible angle for B.
In a triangle, angle B can have another possible value, [tex]\( B'\)[/tex], where:
[tex]\[ B' = 180^\circ - B \][/tex]
[tex]\[ B' = 180^\circ - 63^\circ = 117^\circ \][/tex]
Step 6: Check the possible values for angle C.
For the first scenario:
[tex]\[ C = 180^\circ - ZA - B \][/tex]
[tex]\[ C = 180^\circ - 48^\circ - 63^\circ = 69^\circ \][/tex]
For the second scenario:
[tex]\[ C' = 180^\circ - ZA - B' \][/tex]
[tex]\[ C' = 180^\circ - 48^\circ - 117^\circ = 15^\circ \][/tex]
Step 7: Determine the number of valid triangles.
Both [tex]\( C \ (\approx 69^\circ) \)[/tex] and [tex]\( C' \ (\approx 15^\circ) \)[/tex] are valid angles as they are both positive and less than 180°.
Conclusion:
Since there are two valid sets of angles that satisfy the triangle conditions, there are two possible triangles in this scenario. Hence:
There are two possible triangles based on the given parameters.
