Law of Sines

The lesson introduces the Law of Sines for solving missing sides and angles in triangles. It begins by reviewing different triangle types (acute, obtuse, right, equiangular) and focuses on oblique triangles—those without a right angle. Oblique triangles can be acute (all angles < 90°) or obtuse (one angle > 90°). Two key restrictions for oblique triangles are: the sum of any two angles must be less than 180°, and the sum of the two shorter sides must be greater than the longest side. The lesson also covers how to classify triangles based on given angles and sides, such as ASA, AAS, SSA, SSS, and SAS.

The Law of Sines, expressed as a/sin A = b/sin B = c/sin C, is applicable for oblique triangles when given two angles and any side (AAS or ASA). The lesson demonstrates solving such triangles. For example, given two angles and a non-included side, the third angle can be found by subtracting the sum of the known angles from 180°. Then, the Law of Sines is used to find the missing sides by setting up proportions. The concept emphasizes choosing two parts of the Law of Sines formula where only one variable is unknown at a time.

The lesson moves on to the ambiguous case (SSA), where two sides and an angle opposite one of them are given. This case is 'ambiguous' because there can be zero, one, or two possible triangles. The approach depends on whether the given angle is acute, obtuse, or right. For an acute angle, the height (h = side_not_opposite * sin(angle)) is calculated to determine the number of triangles. Comparisons between the side opposite the given angle ('a'), the height ('h'), and the other given side ('b') dictate the outcome.

The lesson elaborates on the possible outcomes for SSA: - No triangle: If the side opposite the angle ('a') is shorter than the height ('h'). - Two triangles: If 'a' is longer than 'h' but shorter than the other given side ('b'). - One right triangle: If 'a' equals 'h'. - One triangle: If 'a' is greater than 'b' (and therefore also greater than 'h'). - One isosceles triangle: If 'a' equals 'b' and is greater than 'h'. For obtuse or right given angles, the height calculation is not needed; a simple comparison between 'a' and 'b' (side opposite angle vs. other side) determines if one or no triangle can be formed.

The lesson provides practical examples of solving the SSA case. One example starts with an obtuse angle, clarifying that only one triangle can be formed if the side opposite the obtuse angle is longer than the other given side. Another challenge involves an acute angle, where calculating the height reveals that two triangles can be formed. The process involves using the Law of Sines to find a missing angle, then calculating the third angle, and finally using the Law of Sines again to find the remaining side for each possible triangle.