Law of Sines, Basic Introduction, AAS & SSA - One Solution, Two Solutions vs No Solution, Trigonomet

Summary

This video provides a basic introduction to the Law of Sines, explaining its formula and demonstrating how to use it to solve triangles. It covers scenarios with one solution, two solutions (ambiguous case), and no solution, using various examples.

Highlights

Understanding the Law of Sines Formula00:00:01

The Law of Sines is introduced with the formula a/sin A = b/sin B = c/sin C, where lowercase letters represent side lengths and uppercase letters represent angles. This law is used to find missing angles or sides in a triangle, remembering that the sum of angles in a triangle is 180 degrees.

Example 1: Solving a Triangle (AAS Case - One Solution)00:01:08

Given angle A (60 degrees), angle B (70 degrees), and side a (8), the first step is to calculate angle C (180 - 60 - 70 = 50 degrees). Then, the Law of Sines is applied to find side b (8.68) and side c (7.07), ensuring the calculator is in degree mode. The lengths are verified to correspond with their opposite angle sizes.

Example 2: Analyzing SSA - The Ambiguous Case (One Solution)00:05:31

Given angle A (42 degrees), side a (10), and side b (9), this is an SSA (Side-Side-Angle) case. First, angle B is calculated using the Law of Sines, resulting in 37.03 degrees. To check for a second solution, 180 - 37.03 = 142.97. Adding this to angle A (42 + 142.97 = 184.97) exceeds 180, indicating only one triangle is possible. Angle C and side C are then calculated.

Example 3: SSA Case - No Solution00:11:44

Given angle A (75 degrees), side a (8), and side c (9). Using the Law of Sines to find angle C, the calculation leads to sin C = 1.089. Since the sine of an angle cannot be greater than 1, this means there is no possible triangle that fits these dimensions, hence no solution.

Example 4: SSA Case - Two Solutions00:14:02

Given angle A (30 degrees), side a (7), and side b (8). Solving for angle B first yields 34.85 degrees. Checking for a second solution: 180 - 34.85 = 145.15 degrees. Adding this to angle A (30 + 145.15 = 175.15) which is less than 180, indicates two possible triangles. Both triangles are solved, calculating angle C and side c for each, and verifying the side-angle consistency.