Section 4.2 – The Law of Sines

If none of the angles of a triangle is a right angle, the triangle is called oblique. An oblique triangle has either three acute angles or two acute angles and one obtuse angle. To solve an oblique triangle means to find the lengths of its sides and the measurements of its angles..

When solving oblique triangles, there are four cases to consider: 1.One side and two angles are known (ASA or SAA). 2.Two sides and the angle opposite them are known (SSA). 3.Two sides and the included angle are known (SAS). 4.Three sides are known (SSS).

Cases 1 and 2 are solved using the Law of Sines 1.One side and two angles are known (ASA or SAA). 2.Two sides and the angle opposite them are known (SSA).

A A S CASE 1: ASA

AA S CASE 1: SAA

S A S CASE 2: SSA

Theorem Law of Sines

Remember The sum of the angle measurements in a triangle is 180 degrees. α + β + γ = 180 degrees

Solve the SAA triangle where α = 40 o, β = 60 o, and a = o 40 o α + β + γ = 180 o 40 o + 60 o + γ = 180 o γ = 80 o

4 60 o 40 o

Case 2 (SSA) is called the ambiguous case because it can result in no triangle, one triangle, or two triangles. With two sides a and b known, and the angle α known, we have four cases: 1.If a < b sinα then there is no triangle. 2.If a = b sinα then we have one right triangle. 3.If a b sinα the we have two distinct triangles that can be formed. 4.If a ≥ b then we have only one triangle.

Determine the number of triangles we have when solving the triangle with a = 6, b = 8, and α = 35 o We have a < b. b sin α = 8 sin35 o ≈ 4.59 So, a > b sin α We will have two triangles with the given measurements.

35 o TWO TRIANGLE SOLUTIONS