Summary
Highlights
The video introduces the Law of Sines and focuses on the SSA (Side-Side-Angle) case, also known as the ambiguous case. It explains that two triangles can be created if the opposite side of the given angle is less than the side adjacent to the angle.
The video illustrates a scenario where the opposite side of the given angle is greater than the side adjacent to the angle. In this situation, only one triangle can be formed because rotating the side would extend beyond the base or create an invalid triangle.
The ambiguous case arises when the side opposite the given angle is less than the adjacent side. This allows for two possible triangles to be formed by rotating the opposite side. The video visually demonstrates how this rotation creates a second triangle with the same side lengths and angle A but a different angle B.
The two possible triangles are separated and labeled. The original triangle and the newly formed 'small' triangle share angle A (27 degrees), side a (15), and side b (20). The goal is to find all missing angles and sides for both triangles.
The sine law (sin A / a = sin B / b) is used to find angle B in the first triangle. Substituting the known values (sin 27 / 15 = sin B / 20) and solving for sin B, then taking the inverse sine, gives angle B approximately 37.25 degrees.
The video explains that the rotation creates an isosceles triangle within the larger figure, meaning two sides are equal, and consequently, two base angles are congruent. This relationship is used to determine that the supplementary angle to the calculated angle B (37.25 degrees) forms the angle B of the second, smaller triangle (180 - 37.25 = 142.75 degrees).
Since the sum of angles in any triangle is 180 degrees, Angle C for the first triangle is calculated by subtracting the sum of Angle A (27 degrees) and Angle B (37.25 degrees) from 180 degrees, resulting in 115.75 degrees.
Similarly, Angle C for the second triangle is found by subtracting the sum of Angle A (27 degrees) and the second Angle B (142.75 degrees) from 180 degrees, which gives 10.25 degrees.
Using the sine law (sin C / c = sin A / a), Side C for the first triangle is calculated. Plugging in the known values (sin 115.75 / c = sin 27 / 15) and solving for c gives approximately 29.76.
Finally, Side C for the second triangle is calculated using the sine law again (sin C / c = sin A / a). Using the values for the second triangle (sin 10.25 / c = sin 27 / 15) results in Side C being approximately 5.88.