Law of Sines What You will learn: Use the Law of Sines to solve oblique triangles when you know two angles and one side (AAS or ASA). Use the Law of Sines to solve oblique triangles when you know two sides and the angle opposite one of them (SSA).

Given Two Angles and One Side – AAS For the triangle below C = 102, B = 29, and b = 28 feet. Find the remaining angle and sides. By the triangle angle-sum theorem, A =

Law of Sines Try this: A B C c a b By the triangle angle-sum theorem, C = 𝑎= 𝑐 𝑠𝑖𝑛𝐶 𝑠𝑖𝑛𝐴 𝑏= 𝑐 𝑠𝑖𝑛𝐶 𝑠𝑖𝑛𝐵

Let’s look at this: Example 1 Given a triangle, demonstrate using the Law of Sines that it is a valid triangle (numbers are rounded so they may be up to a tenth off): a = 5 A = 40o b = 7 B = 64.1o c = 7.55 C = 75.9o Is it valid?? a = 5 A = 40o b = 7 B = 115.9o c = 3.175 C = 24.1o Is it valid??

The Ambiguous Case (SSA)

Example – Single-Solution Case—SSA In triangle ABC, a = 12 inches, b = 5 inches, and A = 31. Find the remaining side and angles.

Example – No-Solution Case—SSA In triangle ABC, a = 4 inches, b = 14 inches, and A = 60. Find the remaining side and angles.

Example – Two-Solution Case—SSA In triangle ABC, a = 4.5 inches, b = 5 inches, and A = 58. Find the remaining side and angles.

Law of Cosines What You will learn: Use the Law of coSines to solve oblique triangles when you know three sides (SSS). Use the Law of coSines to solve oblique triangles when you know two sides and the included angle (SAS). Always solve for the angle across from the longest side first!

Example – SSS In triangle ABC, a = 6, b = 8, and c = 12. Find the three angles.

Example – SAS In triangle ABC, A = 80, b = 16, and c = 12. Find the remaining side and angles.