Law of Sines and the Ambigous Case

• Law of Sines


  A relationship between the sides and angles of an oblique triangle
• Solving an oblique triangle using the Law of Sines

  1. Find the third angle using the fact that the sum of the interior angles of a triangle is 180 degrees

  2. Apply the Law of Sines formula: sin A/a = sin B/b = sin C/c

  3. Solve for the unknown side(s) by cross-multiplying and rearranging the Law of Sines formula
• The Law of Sines can be written in reciprocal form: a/sin A = b/sin B = c/sin C
• The Law of Sines can take several forms, such as: a = b * sin A / sin B
• When two angles and one side are given in an oblique triangle


  • First find the third angle using the fact that the sum of the interior angles is 180 degrees
  • Then apply the Law of Sines to find the other two sides
• Conditions for the Law of Sines to be applicable

  • One side and two angles are known
  • Two sides and one angle opposite one of those sides are known
• Ambiguous case


  When there are two possible solutions for an oblique triangle using the Law of Sines
• Determining if there is one or two solutions in the ambiguous case

  1. If angle A is greater than 0 degrees but less than 90 degrees:

  2. - If A < b * sin A, then there is no solution

  3. - If A = b * sin A, then there is one solution (a right triangle)

  4. - If A > b * sin A and A < b, then there are two solutions

  5. - If A > b * sin A and A >= b, then there is one solution
• If angle A is greater than 90 degrees but less than 180 degrees, then there are two possible solutions
• Solving triangle ABC using the law of sines

  1. Given information: a = 12 cm, b = 23 cm, angle A = 34 degrees

  2. Find b using law of sine: sin b = (sin 34 * 23) / 12

  3. Since sin b > 1, there is no solution - no triangle can be formed with the given measures
• Solving triangle ABC using the law of sines

  1. Given information: a = 18, angle A = 73 degrees, b = 11

  2. Find sin b = (sin 73 * 11) / 18

  3. 0 < sin b < 1, so there is exactly one triangle that can be formed

  4. Solve for b = 35.76 degrees

  5. Find angle C = 180 - 73 - 35.76 = 71.24 degrees

  6. Find side c = (sin 71.24 * 18) / sin 73 = 17.82 cm
• Solving triangle ABC using the law of sines

  1. Given information: b = 16 cm, angle B = 28 degrees, c = 20 cm

  2. Since b > c * sin B, this is an ambiguous case with two possible solutions

  3. Solution 1: sin C1 = (sin 28 * 20) / 16 = 0.5868, C1 = 35.93 degrees

  4. Angle A = 180 - 28 - 35.93 = 116.07 degrees

  5. Side a = (16 * sin 116.07) / sin 28 = 30.61 cm

  6. Solution 2: sin C2 = (sin 28 * 20) / 16 = 0.5868, C2 = 144.07 degrees

  7. Angle A = 180 - 28 - 144.07 = 7.93 degrees

  8. Side a = (16 * sin 7.93) / sin 28 = 4.7 cm
• There are two possible solutions when given two sides and one angle in a triangle