The Law of Sines
The Law of Sines (also called the Sine Rule) is a fundamental relationship in trigonometry that connects the sides of any triangle to the sines of their opposite angles. Unlike SOH-CAH-TOA, which only works for right triangles, the Law of Sines works for all triangles.
This guide covers the formula and its derivation, when to apply it (AAS, ASA, SSA), how to navigate the ambiguous case, five fully worked examples, real-world applications, and a practice quiz to test your understanding.
1Introduction
Most triangles in the real world are not right triangles. When you need to solve a triangle that has no 90° angle — called an oblique triangle — SOH-CAH-TOA no longer applies directly. This is where the Law of Sines comes in.
The Law of Sines establishes a proportion between each side of a triangle and the sine of the angle opposite that side. This elegant relationship lets you solve for unknown sides and angles when you have the right combination of given information.
Imagine you’re a surveyor measuring the distance across a river. You can’t walk across it, but you can measure angles from two points on your side and the distance between those points. With the Law of Sines, you can calculate the exact distance across the river without ever crossing it.
The Law of Sines is used in surveying, navigation, astronomy, engineering, and physics. Anytime you need to find unknown measurements in a non-right triangle, this law is one of your two primary tools (the other being the Law of Cosines).
2Key Definitions
Oblique Triangle
Any triangle that does not contain a 90° angle. Can be acute (all angles < 90°) or obtuse (one angle > 90°).
Opposite Side & Angle
In standard notation, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.
AAS (Angle-Angle-Side)
Two angles and a non-included side are known. Always yields exactly one triangle.
ASA (Angle-Side-Angle)
Two angles and the included side (between the angles) are known. Always yields exactly one triangle.
SSA (Side-Side-Angle)
Two sides and a non-included angle are known. This is the “ambiguous case” — may yield 0, 1, or 2 triangles.
Ambiguous Case
When SSA information can produce multiple valid triangles, requiring careful analysis of the height.
Included Side / Angle
A side is “included” if it lies between the two known angles. An angle is “included” if it lies between two known sides.
Triangle Angle Sum
The three interior angles of every triangle always sum to exactly 180°.
3The Formula
The Law of Sines can be written in two equivalent forms. The first is useful for finding unknown sides, and the second (the reciprocal) is useful for finding unknown angles.
Law of Sines — Standard Form (for finding sides)
a / sin A = b / sin B = c / sin C
Law of Sines — Reciprocal Form (for finding angles)
sin A / a = sin B / b = sin C / c
Interactive Law of Sines Explorer
Angle C
80.0°
Side b
20.21
Side c
22.98
Adjust the sliders to change the known values. The triangle and computed results update in real time.
In these formulas, lowercase letters (a, b, c) represent side lengths, and uppercase letters (A, B, C) represent the angles opposite those sides respectively.
Why does it work?
The Law of Sines can be derived by dropping an altitude (height) from any vertex to the opposite side and using the definition of sine in the resulting right triangles. Consider triangle ABC with altitude h from vertex C to side c:
In the left right triangle: sin A = h / b → h = b·sin A
In the right right triangle: sin B = h / a → h = a·sin B
Setting them equal: b·sin A = a·sin B
Dividing both sides by sin A·sin B:
a / sin A = b / sin B
Repeating with a different altitude gives us the third ratio, completing the full law.
4When to Use the Law of Sines
The Law of Sines requires you to know at least one side-angle pair (a side and the angle opposite it). It applies to three configurations:
AAS (Angle-Angle-Side)
You know two angles and a side that is not between them.
Strategy: Find the third angle (180° − A − B = C), then use the Law of Sines to find the unknown sides.
Always gives exactly 1 triangle.
ASA (Angle-Side-Angle)
You know two angles and the side between them (the included side).
Strategy: Find the third angle, then use the Law of Sines to find the remaining sides.
Always gives exactly 1 triangle.
SSA (Side-Side-Angle) — The Ambiguous Case
You know two sides and a non-included angle.
Strategy: Use the Law of Sines to find the angle opposite the second known side, then check for ambiguity.
May give 0, 1, or 2 triangles. See Section 5.
If you have SAS or SSS, use the Law of Cosines instead. The Law of Sines needs at least one complete side–angle pair to set up the proportion.
5The Ambiguous Case (SSA)
SSA is called the ambiguous case because given two sides and a non-included angle, the data may be consistent with zero, one, or two different triangles. Understanding why requires thinking about geometry.
Suppose you know angle A, side b (adjacent to A), and side a (opposite A). The key comparison is between side a and the height h = b·sin A:
Case 1: a < h (a < b·sin A) — No Triangle
Side a is too short to reach the opposite side. The arc of radius a from vertex B does not intersect the line through side c. Zero solutions.
Case 2: a = h (a = b·sin A) — One Right Triangle
Side a exactly equals the height, forming a right angle at the foot of the altitude. Exactly one solution (a right triangle).
Case 3: h < a < b — Two Triangles
Side a is long enough to cross the opposite side in two places. The arc of radius a intersects the line in two points on the same side. Two solutions.
Case 4: a ≥ b — One Triangle
When the opposite side is at least as long as the adjacent side, only one triangle is possible. Exactly one solution.
After using the Law of Sines to find sin B, if sin B < 1, check whether the supplement (180° − B) also gives a valid triangle. If A + (180° − B) < 180°, then two triangles exist. Otherwise, only one.
6Worked Examples
Problem: In triangle ABC, A = 40°, B = 60°, and a = 15. Find side b.
Step 1: Find angle C = 180° − 40° − 60° = 80°
Step 2: Set up the Law of Sines: a/sin A = b/sin B
Step 3: 15/sin 40° = b/sin 60°
Step 4: b = 15 · sin 60° / sin 40°
Step 5: b = 15 · 0.8660 / 0.6428
Answer: b ≈ 20.21
Case: AAS — exactly one triangle
Problem: In triangle ABC, A = 50°, C = 70°, and b = 20 (the side between them). Find sides a and c.
Step 1: Find angle B = 180° − 50° − 70° = 60°
Step 2: Set up: a/sin A = b/sin B = c/sin C
Step 3: a/sin 50° = 20/sin 60°
Step 4: a = 20 · sin 50° / sin 60° = 20 · 0.7660 / 0.8660
Step 5: a ≈ 17.69
Step 6: c = 20 · sin 70° / sin 60° = 20 · 0.9397 / 0.8660
Answer: a ≈ 17.69, c ≈ 21.71
Case: ASA — exactly one triangle
Problem: In triangle ABC, A = 50°, b = 10, a = 6. How many triangles exist?
Step 1: Compute height h = b · sin A = 10 · sin 50°
Step 2: h = 10 · 0.7660 = 7.66
Step 3: Compare a to h: a = 6 < 7.66 = h
Step 4: Since a < h, side a is too short to reach the opposite side
Answer: No triangle exists (zero solutions)
Case: SSA — a < h means no triangle
Problem: In triangle ABC, A = 30°, a = 12, b = 8. Solve the triangle.
Step 1: Since a = 12 > b = 8, exactly one triangle (a ≥ b)
Step 2: sin B / b = sin A / a → sin B = 8 · sin 30° / 12
Step 3: sin B = 8 · 0.5 / 12 = 4/12 = 1/3 ≈ 0.3333
Step 4: B = sin¹(0.3333) ≈ 19.47°
Step 5: C = 180° − 30° − 19.47° = 130.53°
Step 6: c = 12 · sin 130.53° / sin 30° = 12 · 0.7604 / 0.5
Answer: B ≈ 19.47°, C ≈ 130.53°, c ≈ 18.25
Case: SSA with a > b — exactly one triangle
Problem: In triangle ABC, A = 30°, a = 7, b = 10. Solve for all possible triangles.
Step 1: Check h = b · sin A = 10 · sin 30° = 10 · 0.5 = 5
Step 2: Since h = 5 < a = 7 < b = 10 → two triangles possible
Step 3: sin B = b · sin A / a = 10 · 0.5 / 7 ≈ 0.7143
Step 4: B₁ = sin¹(0.7143) ≈ 45.58°
Step 5: B₂ = 180° − 45.58° = 134.42°
Triangle 1: B = 45.58°, C = 180° − 30° − 45.58° = 104.42°
c = 7 · sin 104.42° / sin 30° ≈ 13.56
Triangle 2: B = 134.42°, C = 180° − 30° − 134.42° = 15.58°
c = 7 · sin 15.58° / sin 30° ≈ 3.76
Answer: Two valid triangles exist
Case: SSA with h < a < b — two triangles
7Real-World Applications
Surveying & Land Measurement
Surveyors use the Law of Sines to calculate distances across rivers, valleys, or other obstacles by measuring angles from two known points. This technique is called triangulation.
Navigation & Aviation
Pilots and sailors use the Law of Sines to determine position and course corrections when traveling between waypoints that form non-right triangles.
Astronomy
Astronomers apply the Law of Sines to calculate distances to stars using parallax measurements, where the Earth’s orbit forms the baseline of a very large triangle.
Architecture & Engineering
Engineers use the Law of Sines when designing roof trusses, bridges, and other structures that involve non-right-angle supports.
Physics & Force Vectors
When resolving forces that don’t form right angles, the Law of Sines helps determine the magnitude and direction of resultant forces.
8Memory Aids
Remember: side over sine of opposite angle. Each fraction has a side on top and the sine of the angle across from it on the bottom: a/sin A.
The lowercase letter (side) and uppercase letter (angle) in each fraction always match: a with A, b with B, c with C. If they don’t match, something is wrong.
Need a side? Use a/sin A form (side on top). Need an angle? Flip to sin A/a form (sine on top) so you can isolate sin of the unknown angle.
Whenever you see SSA, be suspicious — always check for the ambiguous case. Calculate h = b·sin A first and compare with a.
Law of Sines works for any triangle. SOH-CAH-TOA works only for right triangles. When there’s no 90° angle, reach for the Law of Sines (or Cosines).
9Common Mistakes to Avoid
Forgetting the Ambiguous Case
Wrong: Finding one value of B in SSA and assuming you’re done
Right: Always check if 180° − B also produces a valid triangle when you have SSA.
Mismatching Sides and Angles
Wrong: Writing a/sin B = b/sin A (sides and angles don’t correspond)
Right: Each fraction pairs a side with its opposite angle: a/sin A = b/sin B.
Using Law of Sines for SAS or SSS
Wrong: Trying to apply the Law of Sines when you have SAS (two sides and included angle)
Right: SAS and SSS require the Law of Cosines. The Law of Sines needs a complete side–angle pair.
Not Finding the Third Angle First
Wrong: Jumping into the Law of Sines when you have AAS/ASA without computing the missing angle
Right: When you know two angles, always find the third angle first (C = 180° − A − B) before applying the law.
Calculator in Radian Mode
Wrong: sin(30) = −0.988 (calculator in radian mode)
Right: sin(30°) = 0.5. Always ensure your calculator is in DEGREE mode.
Ignoring sin B > 1
Wrong: Proceeding with a calculation when sin B = 1.2
Right: If sin B > 1, the triangle is impossible. No angle has a sine greater than 1.
10Quick Revision Summary
- Law of Sines: a/sin A = b/sin B = c/sin C.
- Works for any triangle — not just right triangles.
- Use when you have AAS, ASA, or SSA.
- AAS and ASA always give exactly one triangle.
- SSA is the ambiguous case — may give 0, 1, or 2 triangles.
- Compare a with h = b·sin A to determine the number of SSA solutions.
- Use the reciprocal form (sin A/a) when finding angles.
- If sin B > 1, no triangle exists.
- For SAS or SSS, use the Law of Cosines instead.
- Always check your calculator is in DEGREE mode.
- Draw and label the triangle before solving — it prevents most errors.
Frequently Asked Questions
When should I use the Law of Sines vs. Law of Cosines?
Use Law of Sines when you have AAS, ASA, or SSA. Use Law of Cosines when you have SAS or SSS.
How do I handle the ambiguous case?
Compare the given side a with the height h = b·sin A. If a < h: no triangle. If a = h: one right triangle. If h < a < b: two triangles. If a ≥ b: one triangle.
Can the Law of Sines be used for right triangles?
Yes, but SOH-CAH-TOA is usually simpler for right triangles.
What if I get sin B > 1?
If sin B > 1, no triangle exists with those measurements. The given information is inconsistent.
Practice Quiz
Test your understanding — select the correct answer for each question.
1.The Law of Sines states: a/sin A = b/sin B = c/sin C. When is this formula applicable?
2.What is the 'ambiguous case' in the Law of Sines?
3.In triangle ABC, A = 30°, B = 45°, and a = 10. Find b.
4.If in SSA case, a < b·sin A, how many triangles exist?
5.For AAS, which information do you have?
6.In triangle PQR, P = 40°, Q = 60°, and p = 8. What is angle R?
7.In the SSA case, if a > b (where a is opposite A), how many triangles exist?
8.The height h in an SSA triangle (with angle A and sides a, b) is calculated as:
9.Which form of the Law of Sines is useful for finding an angle?
10.In triangle ABC, a = 7, b = 10, A = 30°. Find sin B.
Study Tips
- Master the formula first — be able to write a/sin A = b/sin B = c/sin C from memory before tackling problems.
- Classify the problem — before solving, identify whether you have AAS, ASA, or SSA. This tells you what to expect.
- Always check for ambiguity in SSA — calculate h = b·sin A and compare it with a. This one step prevents the most common errors.
- Practice both forms — solve problems finding missing sides (a/sin A form) AND missing angles (sin A/a form).
- Work through two-triangle cases — the ambiguous case is tested frequently. Practice finding both solutions.