Question

why can the law of sines not be used to solve a triangle if we are given only the lengths of the three sides of the triangle?

choose the correct answer below.

a. to use the law of sines, we must know three angle measures
b. to use the law of sines, we must know the lengths of two sides and the measure of the angle between those sides.
c. to use the law of sines, we must know an angle measure, and the length of the side opposite it.
d. to use the law of sines, we must know an angle measure, the length of the side opposite it, and at least one other angle measure or side length.

Explanation:

The Law of Sines is \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\), where \(a,b,c\) are side lengths and \(A,B,C\) are the angles opposite those sides. If we only know the three side lengths, we have no angle - side pair (an angle measure and the length of the side opposite it) to set up the proportion in the Law of Sines.

• Option A: Knowing three angle measures is not the requirement for the Law of Sines. We need side - angle relationships.
• Option B: Knowing two sides and the included angle is the condition for the Law of Cosines (\(c^{2}=a^{2}+b^{2}-2ab\cos C\)), not the Law of Sines.
• Option C: This is a part of the requirement. But we also need at least one other angle measure or side length to solve the triangle completely using the Law of Sines. For example, if we know \(a\) and \(A\) (\(a\) is the side opposite \(A\)), and then if we know another side \(b\), we can find \(B\) using \(\frac{a}{\sin A}=\frac{b}{\sin B}\); or if we know another angle \(B\), we can find \(b\) using the same formula.
• Option D: This correctly states the requirements for using the Law of Sines. We need an angle - side pair (an angle measure and the length of the side opposite it) and at least one other angle measure or side length.

Answer:

D. To use the law of sines, we must know an angle measure, the length of the side opposite it, and at least one other angle measure or side length.