To solve a triangle, the choice between the law of sines and the law of cosines depends on the given information. For a right triangle where the hypotenuse is sought, both the Pythagorean theorem and the law of cosines (which simplifies accordingly) can be used, reflecting the internal consistency of mathematics.
To determine whether to use the law of sines or the law of cosines for solving a triangle, we must assess the given information. The law of sines is applicable when we are given two angles and one side (AAS or ASA) or two sides and a non-enclosed angle (SSA). The law of cosines is used when we have two sides and the included angle (SAS) or all three sides (SSS).
If you calculate the hypotenuse of a right triangle using the base angles and adjacent sides, the law of cosines simplifies to the Pythagorean theorem, which states that a² + b² = c². Here, you could use either the Pythagorean theorem or the law of cosines, as they will both yield the same result, further emphasizing the consistency of mathematical principles. Should the triangle in question not be a right triangle, then the choice between the two laws would need to be made based on the given information as mentioned previously.