Questions: Use the law of sines to solve the triangle, if possible. A=34.8°, a=28, b=31 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Round side lengths to the nearest whole number and angle measures to one decimal place as needed.) A. There are two possible solutions for the triangle. The measurements for the solution with the smaller angle B are as follows. B₁=□ c₁= □ c₁= □ The measurements for the solution with the larger angle B are as follows. B₂=□° C₂= □ c₂= □ B. There is only one possible solution for the triangle. The measurements for the remaining angles B, C, and side c are as follows. B=□ C= □ c= □ . C. There are no possible solutions for this triangle.

Use the law of sines to solve the triangle, if possible.

A=34.8°, a=28, b=31

Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Round side lengths to the nearest whole number and angle measures to one decimal place as needed.) A. There are two possible solutions for the triangle.

The measurements for the solution with the smaller angle B are as follows.

B₁=□

c₁= □

c₁= □

The measurements for the solution with the larger angle B are as follows.

B₂=□° C₂= □

c₂= □

B. There is only one possible solution for the triangle.

The measurements for the remaining angles B, C, and side c are as follows.

B=□ C= □

c= □ .

C. There are no possible solutions for this triangle.

Transcript text: Use the law of sines to solve the triangle, if possible. \[ A=34.8^{\circ}, a=28, b=31 \] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Round side lengths to the nearest whole number and angle measures to one decimal place as needed.) A. There are two possible solutions for the triangle. The measurements for the solution with the smaller angle $B$ are as follows. \[ \mathrm{B}_{1}=\square \] \[ c_{1}= \] $\square$ \[ c_{1}= \] $\square$ The measurements for the solution with the larger angle $B$ are as follows. \[ \mathrm{B}_{2}=\square^{\circ} \quad \mathrm{C}_{2}= \] $\square$ \[ c_{2}= \] $\square$ B. There is only one possible solution for the triangle. The measurements for the remaining angles B, C, and side c are as follows. \[ B=\square \quad \mathrm{C}= \] $\square$ \[ c= \] $\square$ . C. There are no possible solutions for this triangle.

Solution

Solution Steps

To solve the triangle using the law of sines, we first calculate angle $ B $ using the formula $\frac{\sin A}{a} = \frac{\sin B}{b}$. Once we find angle $ B $, we can determine angle $ C $ since the sum of angles in a triangle is $ 180^\circ $. Finally, we use the law of sines again to find side $ c $ using $\frac{\sin A}{a} = \frac{\sin C}{c}$. We need to check if there are two possible solutions for angle $ B $ (i.e., if $ B $ and $ 180^\circ - B $ are both valid), which would lead to two different triangles.

Step 1: Calculate Angle $ B $

Using the law of sines, we find angle $ B $ as follows: \[ \frac{\sin A}{a} = \frac{\sin B}{b} \] Substituting the known values: \[ \frac{\sin(34.8^\circ)}{28} = \frac{\sin B}{31} \] This gives us two possible angles for $ B $: \[ B_1 = 39.2^\circ \quad \text{and} \quad B_2 = 140.8^\circ \]

Step 2: Calculate Angle $ C $

For each case of $ B $, we can find angle $ C $ using: \[ C = 180^\circ - A - B \] Calculating for both angles: \[ C_1 = 180^\circ - 34.8^\circ - 39.2^\circ = 106.0^\circ \] \[ C_2 = 180^\circ - 34.8^\circ - 140.8^\circ = 4.4^\circ \]

Step 3: Calculate Side $ c $

Using the law of sines again, we can find side $ c $ for both triangles: \[ \frac{\sin A}{a} = \frac{\sin C}{c} \] Calculating for both cases: \[ c_1 = \frac{a \cdot \sin C_1}{\sin A} = \frac{28 \cdot \sin(106.0^\circ)}{\sin(34.8^\circ)} \approx 47 \] \[ c_2 = \frac{a \cdot \sin C_2}{\sin A} = \frac{28 \cdot \sin(4.4^\circ)}{\sin(34.8^\circ)} \approx 4 \]

Final Answer

The measurements for the solution with the smaller angle $ B $ are: \[ B_1 = 39.2^\circ, \quad C_1 = 106.0^\circ, \quad c_1 = 47 \] The measurements for the solution with the larger angle $ B $ are: \[ B_2 = 140.8^\circ, \quad C_2 = 4.4^\circ, \quad c_2 = 4 \] Thus, the answer is A, as there are two possible solutions for the triangle.

\[ \boxed{A} \]

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