To determine the length of the third side of a triangle where two sides and the included angle are known, you can use the Law of Cosines. The Law of Cosines formula is given by:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Here:
- [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of two sides.
- [tex]\( C \)[/tex] is the included angle between these two sides.
- [tex]\( c \)[/tex] is the length of the third side opposite the angle [tex]\( C \)[/tex].
Given:
- [tex]\( a = 33 \)[/tex]
- [tex]\( b = 37 \)[/tex]
- [tex]\( C = 120^{\circ} \)[/tex]
Using the Law of Cosines, the equation to find the length of the third side [tex]\( c \)[/tex] is:
[tex]\[ c^2 = 33^2 + 37^2 - 2 \times 33 \times 37 \times \cos(120^{\circ}) \][/tex]
Therefore, the correct equation to solve is:
[tex]\[ c^2 = 33^2 + 37^2 - 2(33)(37) \cos 120^{\circ} \][/tex]
So, the correct choice is:
[tex]\[ B. \ c^2 = 33^2 + 37^2 - 2(33)(37) \cos 120^{\circ} \][/tex]
This means the correct equation is:
[tex]\[ c^2 = 33^2 + 37^2 - 2(33)(37) \cos 120^{\circ} \][/tex]
