To determine the number of possible triangles with the given side lengths and angle, we can use the Law of Sines. Below are the step-by-step details:
1. Convert the angle [tex]\( A \)[/tex] from degrees to radians:
[tex]\[ A = 68^\circ \][/tex]
We use the fact that [tex]\( 1^\circ = \frac{\pi}{180} \)[/tex] radians.
[tex]\[ A = 68 \times \frac{\pi}{180} \text{ radians} \][/tex]
2. Apply the Law of Sines:
The Law of Sines states:
[tex]\[ \frac{\sin(A)}{a} = \frac{\sin(B)}{b} \][/tex]
Solving for [tex]\( \sin(B) \)[/tex]:
[tex]\[ \sin(B) = \left( \frac{\sin(A)}{a} \right) \times b \][/tex]
3. Substitute the known values [tex]\( a = 4 \)[/tex] and [tex]\( b = 10 \)[/tex]:
[tex]\[ \sin(B) = \left( \frac{\sin(68^\circ)}{4} \right) \times 10 \][/tex]
4. Evaluate [tex]\( \sin(68^\circ) \)[/tex]:
[tex]\[ \sin(68^\circ) \][/tex]
Assuming the sine value for 68 degrees is known.
5. Check the value of [tex]\( \sin(B) \)[/tex] for validity:
The sine function output must be within the range [tex]\([-1, 1]\)[/tex]. If [tex]\(\sin(B)\)[/tex] is outside this range, then no solution exists.
6. Determine possible angles [tex]\( B \)[/tex]:
If [tex]\( \sin(B) \)[/tex] is within the range [tex]\([-1, 1]\)[/tex], there can be two possible values:
[tex]\[ B_1 = \arcsin(\sin(B)) \][/tex]
[tex]\[ B_2 = \pi - B_1 \][/tex]
7. Check if each possible [tex]\( B \)[/tex] results in a valid triangle:
- For [tex]\( B_1 \)[/tex] to form a valid triangle, [tex]\( A + B_1 < \pi \)[/tex]
- For [tex]\( B_2 \)[/tex] to form a valid triangle, [tex]\( A + B_2 < \pi \)[/tex]
8. Conclusion:
We determine whether each case forms a valid triangle. If neither is valid, there are no solutions.
Given all the checks:
The number of solutions is 0.
