Decide on the suitable procedure to solve the triangle given the following information, then solve the triangle. \( \mathrm{a}=26.21, \mathrm{c}=34.75, \mathrm{C}=15^{\circ} 31^{\prime} \) Which of the following procedures should be used to solve the triangle? A. Use the angle sum formula to find the remaining angle, and the law of sines to find the remaining sides. B. Use the law of sines to find an angle. Then use the angle sum formula to find the remaining angle. Use the law of sines to find the remaining side. C. Use the law of cosines to find the third side and the law of sines to find the smaller of the two remaining angles. Use the angle sum formula to find the last angle. D. Use the law of cosines to find the largest angle. Use the law of sines to find either remaining angle. Use the angle sum formula to find the last angle.

To solve the triangle given the information \( a = 26.21 \), \( c = 34.75 \), and \( C = 15^\circ 31' \), we can analyze the situation: 1. We have two sides \( a \) and \( c \), and the included angle \( C \). 2. This situation is suitable for using the Law of Cosines to find the third side \( b \) first, since we have two sides and the included angle. Thus, the most suitable procedure is: **C. Use the law of cosines to find the third side and the law of sines to find the smaller of the two remaining angles. Use the angle sum formula to find the last angle.** ### Step 1: Use the Law of Cosines to find side \( b \) The Law of Cosines states: \[ b^2 = a^2 + c^2 - 2ac \cdot \cos(C) \] Substituting the known values: - \( a = 26.21 \) - \( c = 34.75 \) - \( C = 15^\circ 31' = 15 + \frac{31}{60} = 15.5167^\circ \) First, convert \( C \) to radians for calculation: \[ C \text{ in radians} = 15.5167 \times \frac{\pi}{180} \approx 0.2715 \text{ radians} \] Now, calculate \( b^2 \): \[ b^2 = (26.21)^2 + (34.75)^2 - 2 \cdot (26.21) \cdot (34.75) \cdot \cos(15.5167^\circ) \] Calculating each term: \[ (26.21)^2 \approx 688.4641 \] \[ (34.75)^2 \approx 1209.0625 \] \[ \cos(15.5167^\circ) \approx 0.9659 \] \[ 2 \cdot (26.21) \cdot (34.75) \cdot \cos(15.5167^\circ) \approx 2 \cdot 26.21 \cdot 34.75 \cdot 0.9659 \approx 1745.679 \] Now substituting back: \[ b^2 \approx 688.4641 + 1209.0625 - 1745.679 \approx 151.8476 \] \[ b \approx \sqrt{151.8476} \approx 12.31 \] ### Step 2: Use the Law of Sines to find angle \( A \) Now we can use the Law of Sines: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \] Using \( a \) and \( C \): \[ \frac{26.21}{\sin(A)} = \frac{34.75}{\sin(15.5167^\circ)} \] Calculating \( \sin(15.5167^\circ) \): \[ \sin(15.5167^\circ) \approx 0.2673 \] Now substituting: \[ \frac{26.21}{\sin(A)} = \frac{34.75}{0.2673} \] \[ \sin(A) = \frac{26.21 \cdot 0.2673}{34.75} \approx \frac{7.0205}{34.75} \approx 0.2024 \] Now find \( A \): \[ A \approx \arcsin(0.2024) \approx 11.65^\circ \] ### Step 3: Find angle \( B \) Using the angle sum formula: \[ B = 180^\circ - A - C \] \[ B = 180^\circ - 11.65^\circ - 15.5167^\circ \approx 152.83^\circ \] ### Summary of the triangle - \( A \approx 11.65^\circ \) - \( B \approx 152.83^\circ \) - \( C = 15.5167^\circ \) - \( a = 26.21 \) - \( b \approx 12.31 \) - \( c = 34.75 \) Thus, the triangle is solved with the angles and sides calculated.