Questions: Solve the triangle with the given parts. a=1080 in b=869 in c=881 in Round as directed by the accompanying table. Select the correct choice below and, if necessary, fill in the answer boxes within the choice. A. There is only one possible solution for the triangle. The measurements for the angles A, B, and C are as follows. A= B= ° C= ° (Type your answers using the appropriate number of significant digits.) B. There are two possible solutions for the triangle. The measurements for the solution with the larger angle A are as follows. A1= ° B1= ° c1= ° The measurements for the solution with the smaller angle A are as follows. A2= ° B2= ° C2= ° (Type your answers using the appropriate number of significant digits.) C. There are no possible solutions for the triangle.

Solve the triangle with the given parts. a=1080 in b=869 in c=881 in Round as directed by the accompanying table. Select the correct choice below and, if necessary, fill in the answer boxes within the choice. A. There is only one possible solution for the triangle. The measurements for the angles A, B, and C are as follows. A= B= ° C= ° (Type your answers using the appropriate number of significant digits.) B. There are two possible solutions for the triangle. The measurements for the solution with the larger angle A are as follows. A1= ° B1= ° c1= ° The measurements for the solution with the smaller angle A are as follows. A2= ° B2= ° C2= ° (Type your answers using the appropriate number of significant digits.) C. There are no possible solutions for the triangle.
Transcript text: Solve the triangle with the given parts. \[ \mathrm{a}=1080 \text { in } b=869 \text { in } c=881 \text { in } \] Round as directed by the accompanying table. Select the correct choice below and, if necessary, fill in the answer boxes within the choice. A. There is only one possible solution for the triangle. The measurements for the angles $\mathrm{A}, \mathrm{B}$, and C are as follows. $A=$ $\square$ \[ \mathrm{B}=\square^{\circ} \] $\square$ \[ \mathrm{C}=\square^{\circ} \] $\square$ (Type your answers using the appropriate number of significant digits.) B. There are two possible solutions for the triangle. The measurements for the solution with the larger angle $A$ are as follows. \[ \mathrm{A}_{1}=\square^{\circ} \] $\square$ \[ \mathrm{B}_{1}=\square^{\circ} \] $\square$ \[ c_{1}=\square^{\circ} \] $\square$ The measurements for the solution with the smaller angle $A$ are as follows. \[ \mathrm{A}_{2}=\square^{\circ} \quad \mathrm{B}_{2}=\square^{\circ} \] $\square$ \[ \mathrm{C}_{2}=\square^{\circ} \] $\square$ (Type your answers using the appropriate number of significant digits.) C. There are no possible solutions for the triangle.
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Solution

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Solution Steps

To solve the triangle with given sides, we can use the Law of Cosines to find one of the angles, and then use the Law of Sines or the Law of Cosines again to find the other angles. The Law of Cosines is useful for finding an angle when all three sides are known. Once one angle is found, the Law of Sines can be used to find another angle, and the third angle can be found by subtracting the sum of the two known angles from 180 degrees.

Step 1: Calculate Angle \( A \)

Using the Law of Cosines, we find angle \( A \) as follows: \[ A = \cos^{-1}\left(\frac{b^2 + c^2 - a^2}{2bc}\right) = \cos^{-1}\left(\frac{869^2 + 881^2 - 1080^2}{2 \cdot 869 \cdot 881}\right) \approx 76.21^\circ \]

Step 2: Calculate Angle \( B \)

Next, we apply the Law of Sines to find angle \( B \): \[ B = \sin^{-1}\left(\frac{b \cdot \sin(A)}{a}\right) = \sin^{-1}\left(\frac{869 \cdot \sin(76.21^\circ)}{1080}\right) \approx 51.39^\circ \]

Step 3: Calculate Angle \( C \)

Finally, we can find angle \( C \) using the fact that the sum of angles in a triangle is \( 180^\circ \): \[ C = 180^\circ - A - B \approx 180^\circ - 76.21^\circ - 51.39^\circ \approx 52.39^\circ \]

Final Answer

The measurements for the angles are: \[ A \approx 76.21^\circ, \quad B \approx 51.39^\circ, \quad C \approx 52.39^\circ \] Thus, the answer is \( \boxed{A} \).

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