Law of Sines- Complete Reference Table and Practice Problems
What Is the Law of Sines?
The Law of Sines is a trigonometric relationship that connects the sides of any triangle to the sines of its angles. It's the tool you reach for when you have partial triangle information and need to find missing pieces.
The formula is straightforward:
a/sin(A) = b/sin(B) = c/sin(C)
Where a, b, c are the side lengths and A, B, C are their opposite angles. You can use any two ratios—the third is just confirmation.
When to Use the Law of Sines
This law works in specific situations:
- SSA (Side-Side-Angle): You know two sides and an angle that isn't between them
- You need to find an unknown angle or side
- At least one angle-side pair is complete
⚠️ Warning: SSA can produce the ambiguous case. Sometimes you'll get two possible triangles. We'll cover this below.
Law of Sines Complete Reference Table
| Scenario | Given Information | Law of Sines Use |
|---|---|---|
| Find missing side | Two angles + one side | Set up ratio with known side, solve for unknown |
| Find missing angle | Two sides + one angle (not included) | Set up ratio, use inverse sine |
| Ambiguous case | Two sides + angle opposite one known side | Check if h = a·sin(B) comparison yields 0, 1, or 2 solutions |
| Area calculation | Two sides + included angle | Use: Area = ½ab·sin(C) |
The Ambiguous Case Explained
When you know side a, side b, and angle A (but A is not between a and b), you might have zero, one, or two valid triangles. Here's how to determine which:
Step 1: Calculate the Height
h = b · sin(A)
Step 2: Compare a to h
- If a < h → No triangle exists
- If a = h → Exactly one right triangle
- If h < a < b → Two possible triangles
- If a ≥ b → Exactly one triangle
This is why the Law of Sines isn't always a clean solve. The ambiguous case trips up most students.
How to Apply the Law of Sines
Example Problem
Given: Angle A = 35°, side a = 7, side b = 10
Find: Angle B
Step 1: Set up the ratio
7/sin(35°) = 10/sin(B)
Step 2: Cross-multiply
7 · sin(B) = 10 · sin(35°)
Step 3: Solve for sin(B)
sin(B) = (10 · 0.574) / 7
sin(B) = 5.74 / 7
sin(B) = 0.82
Step 4: Find the angle
B = sin⁻¹(0.82) ≈ 55°
Step 5: Check for the ambiguous case
h = 10 · sin(35°) ≈ 5.74
Since 7 > 5.74 and 7 < 10, there could be a second solution at 180° - 55° = 125°.
Practice Problems
Problem 1:
Triangle ABC has A = 50°, a = 12, b = 15. Find angle B.
Problem 2:
Triangle ABC has A = 30°, B = 70°, and a = 8. Find side b.
Problem 3:
Triangle ABC has a = 6, b = 9, and A = 25°. Determine how many triangles are possible.
Solutions
Problem 1:
sin(B)/15 = sin(50°)/12
sin(B) = 15 · 0.766 / 12 = 0.9575
B ≈ 73° (also potentially 107° since a < b)
Problem 2:
First find C: 180° - 50° - 70° = 60°
8/sin(50°) = b/sin(70°)
b = 8 · 0.940 / 0.766 ≈ 9.8
Problem 3:
h = 9 · sin(25°) ≈ 3.8
Since 3.8 < 6 < 9, two triangles are possible.
Law of Sines vs. Law of Cosines
| Feature | Law of Sines | Law of Cosines |
|---|---|---|
| Best for | SSA, ASA, AAS cases | SAS, SSS cases |
| Known info | At least one angle-side pair | Two sides + included angle, or all three sides |
| Ambiguous case | Yes (SSA) | No |
| Formula | a/sin(A) = b/sin(B) = c/sin(C) | c² = a² + b² - 2ab·cos(C) |
Use Law of Sines when you have an angle and its opposite side. Use Law of Cosines when the angle is夹在 between two known sides (SAS) or when you only know three sides (SSS).
Common Mistakes to Avoid
- Forgetting to check the ambiguous case when working with SSA
- Using degrees when your calculator is in radians (or vice versa)
- Rounding too early in calculations—keep more decimals until the final answer
- Confusing which side goes with which angle (side a is always opposite angle A)
- Forgetting that sin(θ) = sin(180° - θ) in the ambiguous case
Quick Reference Cheat Sheet
Formula: a/sin(A) = b/sin(B) = c/sin(C)
When solving for an angle:
sin(B) = (b · sin(A)) / a
B = sin⁻¹(result)
When solving for a side:
b = (sin(B) · a) / sin(A)
For area:
Area = ½ · side₁ · side₂ · sin(included angle)
That's the Law of Sines. The formula is simple. The execution is where people lose points. Pay attention to the ambiguous case and always verify your angle-side pairings.