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

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.

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.

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.

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.

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.