CMClearMathAcademy

Law of Sines

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

The law of sines connects the sine of each angle of a triangle to the length of the opposite side: sin(A)/a = sin(B)/b = sin(C)/c. It works for any triangle, not just right triangles. Use it when you know two angles and a side (AAS or ASA), or two sides and a non-included angle (SSA). In Triangles, Similarity, and Trigonometry, students need to read the diagram, name the relationship, choose a theorem or formula, and justify why the result follows. The expanded practice now includes fluency, transfer, cumulative review, and proof-style reasoning so Geometry feels connected instead of isolated by topic.

What you'll learn

Why it matters: Surveying, navigation, and astronomy all use the law of sines to compute distances and angles to inaccessible points using one known side and two known angles.

Worked example

Problem. In triangle ABC, angle A = 30 degrees, angle B = 45 degrees, and side a = 10. Find side b.

  1. Law of sines: sin(A)/a = sin(B)/b.
  2. sin(30)/10 = sin(45)/b.
  3. (1/2)/10 = (sqrt(2)/2)/b, so b = 10 * sqrt(2).
  4. Connect the result back to Law of Sines so the geometric relationship is explicit.

Answer: 10 * sqrt(2)

Practice problems

1. Law of sines pairs sin(A) with which side?

Choices: The side opposite angle A · The side next to A · The longest side

Show solution
  1. Warm-up: First identify exactly what the question is asking: Law of sines pairs sin(A) with which side?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Opposite-angle and opposite-side go together in the law of sines.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

Answer: The side opposite angle A

2. Triangle with A = 30, B = 60, a = 5. Find sin(A)/a as a fraction.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Triangle with A = 30, B = 60, a = 5. Find sin(A)/a as a fraction.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. sin(30) = 1/2.
  4. (1/2)/5 = 1/10.
  5. Check the result by substituting or estimating: the response should match 1/10 and make sense in the original problem.

Answer: 1/10

3. Triangle with angles A = 30, B = 45 (so C = 105) and a = 6. Find b (the side opposite B) using sin(45) = sqrt(2)/2. Enter the coefficient in front of sqrt(2).

Show solution
  1. sin(30)/6 = sin(45)/b.
  2. (1/2)/6 = (sqrt(2)/2)/b.
  3. Cross multiply: b/2 = 6 * sqrt(2)/2 = 3*sqrt(2)... wait let me redo: b * (1/2) / 6 = sqrt(2)/2, so b * (1/12) = sqrt(2)/2, b = 12 * sqrt(2)/2 = 6 * sqrt(2). Coefficient is 6.
  4. Identify the diagram relationship, formula, theorem, or proof reason before calculating.
  5. Use the given measurements, coordinates, or congruent parts to set up the geometry statement.

Answer: 6

4. Triangle with A = 45 and B = 60 and side a = 4. Find side b. Enter the coefficient in front of sqrt(6).

Show solution
  1. sin(45)/4 = sin(60)/b.
  2. (sqrt(2)/2)/4 = (sqrt(3)/2)/b.
  3. Cross-multiply and simplify: b = 4 * sqrt(3) / sqrt(2) = 4 * sqrt(3) * sqrt(2) / 2 = 2 * sqrt(6). Coefficient = 2.
  4. Identify the diagram relationship, formula, theorem, or proof reason before calculating.
  5. Use the given measurements, coordinates, or congruent parts to set up the geometry statement.

Answer: 2

5. If sin(A)/a = sin(B)/b, and a = 8, b = 12, sin(A) = 1/2, find sin(B).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If sin(A)/a = sin(B)/b, and a = 8, b = 12, sin(A) = 1/2, find sin(B).
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. (1/2)/8 = sin(B)/12.
  4. sin(B) = 12 * (1/2)/8 = 6/8 = 3/4.
  5. Check the result by substituting or estimating: the response should match 3/4 and make sense in the original problem.

Answer: 3/4

Practice this interactively with instant feedback and an AI tutor.

Practice Law of Sines Take the free placement check

More Geometry lessons