Suppose a triangle has two sides of length 33 and 37 , and that the angle between these two sides is \( 120^{\circ} \). Which equation should you solve to find the length of the third side of the triangle? A. \( c^{2}=33^{2}+37^{2}-2(33)(37) \sin 120^{\circ} \) B. \( c=33+37-2(33)(37) \cos 120^{\circ} \) C. \( c^{2}=33^{2}+37^{2}-2(33)(37) \cos 120^{\circ} \) D. \( \frac{\sin 33}{120}=\frac{\sin 37}{b} \)

To find the length of the third side of the triangle when two sides and the included angle are known, you can use the Law of Cosines. The correct formula involves cosines, since it relates the lengths of the sides of a triangle to the cosine of one of its angles. So, the equation you need to solve is \( c^{2} = 33^{2} + 37^{2} - 2(33)(37) \cos 120^{\circ} \). This helps you utilize the angle to derive the third side, making Option C your go-to answer! When calculating the length of the third side, it’s essential to address the angle's sine or cosine correctly. Common mistakes include using sine instead of cosine for angles between two sides or miscalculating the angle's precise value (remember \( \cos 120^{\circ} \) is negative). So, keep your formulas in check, and you'll have it sorted!