Laws of Sines

Introduction  In the last module we studied techniques for solving RIGHT triangles.  In this section and the next, you will solve OBLIQUE TRIANGLES – triangles that have no right angles.

Introduction Cont.  As standard notation,  The angles of a triangle are labeled A, B, and C, and their  Opposite sides are labeled a, b, and c.

Solving Oblique Triangles  To solve an OBLIQUE Triangle, you need to kown the measure of  At least one side and  Any two other parts of the triangle (either two sides, two angles, or one angle and one side)  This breaks down into the following four cases. 1. Two angles and any side (AAS or ASA) 2. Two sides and an angle opposite one of them (SSA) 3. Three sides (SSS) 4. Two sides and their included angles (SAS)  The first two cases can be solved using the LAW of SINES, whereas the last two cases require the LAW of COSINES. Law of Sines Law of Cosines

Law of Sines C is Acute C is Obtuse

AAS  Given Two Angles and One Side  We know two angles and a side.  We can find the third angle by adding the two known angles and subtracting from 180 o.  Once we have all three angles we can use the Law of Sines to find the unknown sides.

ASA  Given Two Angles and One Side  We know two angles and the side that lies between them. We can find the third angle by adding the two known angles and subtracting from 180 o. Once we have all three angles we can use the Law of Sines to find the unknown sides

Law of Sines (The Ambiguous Case – SSA)  Last class, we learned how to apply the Laws of Sines if given two angles and one side (AAS & ASA).  However…if two sides and one opposite angle are given, then three possible situations can occur. 1. No such triangle exists 2. One such triangle exists or 3. Two distinct triangles may satisfy the conditions.  Add the diagram as an attachment to your Cornell Notes.

Law of Sines (The Ambiguous Case – SSA)  Consider a triangle in which you are given a, b, and A. (h = b sin A)

SSA  Given  We know an angle and two sides. This is frequently called the ambiguous case We can use the Law of Sines to try to find the second angle. We may find no solutions, one solution or two solutions.  Let’s discuss the Examples in the book  Example 3: One-Solution Case  Example 4: No-Solution Case  Example 5: Two-Solution Case

Assessment  Determine the number of triangles possible in each of the following cases.  A = 62º, a = 10, b = 12  A = 98º, a = 10, b = 3