In order to apply the law of cosines to find the length of the side of a triangle,...
Check the final answer first, then review the worked steps.
Problem
In order to apply the law of cosines to find the length of the side of a triangle, it is enough to know which of the following?
Answer
B
Step-by-step solution
- Understand the Law of Cosines: The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is typically expressed as $c^2 = a^2 + b^2 - 2ab \cos(C)$, where $a$, $b$, and $c$ are the lengths of the sides of the triangle, and $C$ is the angle opposite side $c$. This formula can be rearranged to solve for any side if the other two sides and the included angle are known, or to solve for an angle if all three sides are known.
- Analyze Option A (Area of the triangle): Knowing only the area of a triangle is not sufficient to apply the Law of Cosines to find the length of a side. The area formula (e.g., Area = $\frac{1}{2}ab \sin(C)$) involves side lengths and an angle, but knowing only the area does not uniquely determine the side lengths or angles needed for the Law of Cosines.
- Analyze Option B (Two sides and the included angle): If you know the lengths of two sides (e.g., $a$ and $b$) and the measure of the angle between them (e.g., $C$), you can directly use the Law of Cosines to find the length of the third side ($c$). The formula $c^2 = a^2 + b^2 - 2ab \cos(C)$ directly uses these known values to calculate $c^2$, and thus $c$.
- Analyze Option C (Two angles and one side): Knowing two angles and one side allows you to find the third angle (since the sum of angles in a triangle is $180^\circ$). With two angles and one side, you can use the Law of Sines to find the lengths of the other two sides. However, the Law of Cosines is not directly applied in this scenario to find a side length. While you could eventually find all sides using the Law of Sines, the question specifically asks what is needed to apply the law of cosines to find a side length. The Law of Sines is more direct for this case (Angle-Angle-Side or AAS, and Angle-Side-Angle or ASA). The Law of Cosines is most directly applied in the Side-Angle-Side (SAS) case to find the third side, or the Side-Side-Side (SSS) case to find an angle.
- Conclusion: To directly apply the Law of Cosines to find the length of a side, you need to know the lengths of the other two sides and the measure of the angle between them (SAS case).