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 Cosine

s

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 180o. 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 180o. 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 exists2. One such triangle exists or3. 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