Find the unknown angles in triangle $\mathrm{ABC}$ for each triangle that exists.
\[
A=75.9^{\circ} \quad b=9.6 \mathrm{ft} \quad a=12.4 \mathrm{ft}
\]
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Round to the nearest tenth as needed.)
A. There are two possible solutions for the triangle. The measurements for when $B$ is larger are $B=\square^{\circ}$ and $C=\square^{\circ}$. The measurements for when $B$ is smaller are $B=\square^{\circ}$ and $C=\square^{\circ}$.
B. There is only one possible solution for the triangle. The measurements for the remaining angles are $B=\square^{\circ}$ and $\mathrm{C}=$
C. There are no possible solutions for the triangle.
Answer
Final Answer: The correct choice is B. There is only one possible solution for the triangle. The measurements for the remaining angles are \(B=48.7^\circ\) and \(C=55.4^\circ\). \(\boxed{B=48.7^\circ, C=55.4^\circ}\)
Step 1 ：Given that we know the values of angle A and sides a and b, we can use the Law of Sines to find the possible values for angle B. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides of the triangle. This can be written as: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]
Step 2 ：First, we can rearrange the Law of Sines to solve for B: \[B = \sin^{-1}\left(\frac{b \sin A}{a}\right)\]
Step 3 ：Then, we can calculate the value of B. However, since the sine function has a period of 180 degrees, there are two possible solutions for B: B and 180 - B.
Step 4 ：Finally, we can find the value of angle C by subtracting the known angles from 180, since the sum of the angles in a triangle is 180 degrees: \[C = 180 - A - B\]
Step 5 ：Let's calculate these values. A = 75.9, a = 12.4, b = 9.6. The calculated values for B and C are 48.7 degrees and 55.4 degrees respectively, and the second possible values for B and C are 131.3 degrees and -27.2 degrees respectively.
Step 6 ：However, the value of an angle in a triangle cannot be negative, so the second solution is not valid. Therefore, there is only one possible solution for the triangle.
Step 7 ：Final Answer: The correct choice is B. There is only one possible solution for the triangle. The measurements for the remaining angles are \(B=48.7^\circ\) and \(C=55.4^\circ\). \(\boxed{B=48.7^\circ, C=55.4^\circ}\)
