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Chapter 5: Problem 62

Determine whether the statement is true or false. Justify your answer. In addition to SSS and SAS, the Law of Cosines can be used to solve triangles with AAS conditions.

Short Answer

Expert verified
False. Typically, the Law of Cosines is used under the SSS (Side-Side-Side) and SAS (Side-Angle-Side) conditions. For solving triangles in the AAS (Angle-Angle-Side) case, the Law of Sines is usually applied.

Step by step solution

01

Understanding the Basic Trigonometric Laws

There are several ways to solve a triangle, that is, to find the lengths of its sides and the measures of its angles - SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), and AAS (Angle-Angle-Side). In SSS, you know the three sides of the triangle and with SAS you know two sides and the angle between them. The Law of Cosines is typically used in these scenarios when you have the lengths of all sides, or two sides and the included angle.
02

Analyzing where AAS is applicable

The AAS (Angle-Angle-Side) case involves having two angles and a non-included side. This case is typically solved using the Law of Sines and not the Law of Cosines. The Law of Cosines would require either three sides (SSS) or two sides and the included angle (SAS). With AAS, since the side isn't between the two known angles, it's not possible to directly apply the Law of Cosines.
03

Conclusion

Even though it's technically possible to use the Law of Cosines with some manipulations in the AAS case, it's not the straightforward or typical method applied in this scenario. The statement that Law of Cosines can be used to solve triangles with AAS conditions is therefore considered false when considering the typical use cases of these trigonometric laws.

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