Problem 64 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Solving an SSS triangle, I do not have to be concerned about the ambiguous case when using the Law of sines.

Step-by-Step Solution

Answer

Yes, the statement makes sense. When solving a triangle with Side-Side-Side (SSS), you don't need to worry about the ambiguous case while using the Law of sines because, with three given sides, the triangle is uniquely determined.

Step 1: Understanding the Problem

The statement mentions about 'solving an SSS triangle' which means we are given the lengths of all three sides of a triangle. And it says 'using the law of sines' - the law of sines is a formula in trigonometry stating that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides and angles in a given triangle.

Step 2: Analyzing the Law of Sines and the Ambiguous Case

The law of sines can be used to find a missing side or a missing angle in a triangle. The ambiguous case happens when, given two sides and an angle, there are two possible triangles. This is usually denoted as Angle-Side-Side (ASS) or Side-Side-Angle (SSA). It is important to mention that there is no ambiguous case for Side-Side-Side (SSS) because with three given sides, only one triangle is possible.

Step 3: Final Judgment

The statement mentioned in the problem seems to make sense. The ambiguous case can occur while using the Law of sines is not a concern when it comes to a Side-Side-Side (SSS) triangle. The statement makes sense because with three given sides, the triangle is uniquely determined, and there is no possibility of an ambiguous case.