CMClearMathAcademy

Law of Sines

A free Trigonometry lesson from the “Applications of Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.

The Law of Sines states that in any triangle, each side divided by the sine of its opposite angle gives the same ratio: a/sinA = b/sinB = c/sinC. You use it when you know a matched side-angle pair plus one more piece of information, such as in AAS or ASA cases. For ASA, first subtract the two known angles from 180 degrees to find the third angle before applying the ratio.

What you'll learn

Why it matters: Surveyors and navigators use the Law of Sines to find inaccessible distances, like the width of a river or the distance to a landmark, by measuring two angles and one baseline length.

Worked example

Problem. In triangle ABC, angle A = 40 degrees, angle B = 75 degrees, and side a = 12. Find side b to the nearest tenth.

  1. Set up the ratio: a/sinA = b/sinB, so 12/sin(40) = b/sin(75).
  2. Solve for b: b = 12 * sin(75) / sin(40).
  3. Compute: b = 12 * 0.9659 / 0.6428 = 18.0.

Answer: 18.0

Practice problems

1. In triangle ABC, angle A = 30 degrees, angle B = 70 degrees, and side a = 8. Find side b to the nearest tenth.

Show solution
  1. Warm-up: First identify exactly what the question is asking: In triangle ABC, angle A = 30 degrees, angle B = 70 degrees, and side a = 8. Find side b to the nearest tenth.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Set up 8/sin(30) = b/sin(70).
  4. Solve: b = 8 * sin(70) / sin(30).
  5. Compute: b = 8 * 0.9397 / 0.5 = 15.0.
  6. Check the result by substituting or estimating: the response should match 15.0 and make sense in the original problem.

Answer: 15.0

2. In a triangle, two angles are 50 degrees and 60 degrees. Find the measure of the third angle in degrees.

Show solution
  1. Warm-up: First identify exactly what the question is asking: In a triangle, two angles are 50 degrees and 60 degrees. Find the measure of the third angle in degrees.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. The angles of a triangle sum to 180 degrees.
  4. Third angle = 180 - 50 - 60.
  5. Third angle = 70 degrees.
  6. Check the result by substituting or estimating: the response should match 70 and make sense in the original problem.

Answer: 70

3. In triangle ABC, angle A = 45 degrees, side a = 10, and angle B = 60 degrees. Find side b to the nearest tenth.

Show solution
  1. Warm-up: First identify exactly what the question is asking: In triangle ABC, angle A = 45 degrees, side a = 10, and angle B = 60 degrees. Find side b to the nearest tenth.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Set up 10/sin(45) = b/sin(60).
  4. Solve: b = 10 * sin(60) / sin(45).
  5. Compute: b = 10 * 0.8660 / 0.7071 = 12.2.
  6. Check the result by substituting or estimating: the response should match 12.2 and make sense in the original problem.

Answer: 12.2

4. In triangle ABC, angle A = 35 degrees, angle B = 65 degrees, and the included side c = 20. Find side a to the nearest tenth.

Show solution
  1. Core Practice: First identify exactly what the question is asking: In triangle ABC, angle A = 35 degrees, angle B = 65 degrees, and the included side c = 20. Find side a to the nearest tenth.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Find angle C: 180 - 35 - 65 = 80 degrees.
  4. Set up a/sin(35) = 20/sin(80).
  5. Compute: a = 20 * sin(35) / sin(80) = 11.6.
  6. Check the result by substituting or estimating: the response should match 11.6 and make sense in the original problem.

Answer: 11.6

5. In triangle ABC, angle A = 110 degrees, angle B = 25 degrees, and side a = 30. Find side b to the nearest tenth.

Show solution
  1. Core Practice: First identify exactly what the question is asking: In triangle ABC, angle A = 110 degrees, angle B = 25 degrees, and side a = 30. Find side b to the nearest tenth.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Set up 30/sin(110) = b/sin(25).
  4. Solve: b = 30 * sin(25) / sin(110).
  5. Compute: b = 30 * 0.4226 / 0.9397 = 13.5.
  6. Check the result by substituting or estimating: the response should match 13.5 and make sense in the original problem.

Answer: 13.5

Practice this interactively with instant feedback and an AI tutor.

Practice Law of Sines Take the free placement check

More Trigonometry lessons