You’ve mastered right triangles with SOH CAH TOA, and you’re feeling confident. Then your teacher throws a triangle at you that doesn’t have a 90° angle, and suddenly everything you know seems useless. Sound familiar? Don’t worry—the Law of Sines is about to become your new best friend for solving these “oblique” triangles.
The Law of Sines is one of the most powerful tools in trigonometry, allowing you to solve any triangle when you have the right combination of angles and sides. In this guide, we’ll break down exactly what the Law of Sines means, when to use it, and how to avoid the tricky pitfalls that trip up so many students.
What Is the Law of Sines? (Plain English, Please!)
At its core, the Law of Sines describes a beautiful relationship that exists in every triangle: the ratio of any side to the sine of its opposite angle is constant throughout the entire triangle.
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C)
Let’s unpack what this actually means. In any triangle:
- Side a is opposite to angle A
- Side b is opposite to angle B
- Side c is opposite to angle C
When you divide any side by the sine of the angle across from it, you get the same number every time. It’s like a secret code built into every triangle!
Why Does This Work?
Think of it this way: bigger angles “open up” to create longer opposite sides. A small angle creates a short opposite side; a large angle creates a long opposite side. The Law of Sines captures this proportional relationship mathematically.
This constant ratio is actually equal to the diameter of the circumscribed circle (the circle that passes through all three vertices of the triangle). Pretty cool, right?
When Should You Use the Law of Sines?
Here’s the crucial question every student asks: “How do I know when to use the Law of Sines instead of the Law of Cosines?”
The Law of Sines works best when you have information about angle-side pairs—meaning you know both an angle and the side opposite to it. Specifically, it handles three scenarios:
The Three Cases for Law of Sines
| Case | What You Know | Example |
|---|---|---|
| ASA | Two angles and the included side | Angles A and B, and side c between them |
| AAS | Two angles and a non-included side | Angles A and B, and side a (not between them) |
| SSA | Two sides and an angle opposite one of them | Sides a and b, and angle A opposite side a |
When to Use Law of Sines
Use Law of Sines when: You have a matching pair (an angle and its opposite side).
Use Law of Cosines when: You have SAS (two sides and the included angle) or SSS (all three sides).
Step-by-Step Examples for Each Case
Let’s work through each case type so you can see exactly how to apply the Law of Sines.
Example 1: AAS Case (Two Angles and a Non-Included Side)
Problem: In triangle ABC, angle A = 42°, angle B = 73°, and side a = 12 cm. Find side b.
Step 1: Identify what you have and what you need.
- Known: A = 42°, B = 73°, a = 12
- Find: b
Step 2: Set up the Law of Sines proportion.
Since we know angle A and side a (an opposite pair), and we want to find side b (opposite angle B):
a/sin(A) = b/sin(B)
Step 3: Substitute the known values.
12/sin(42°) = b/sin(73°)
Step 4: Solve for b.
b = 12 × sin(73°)/sin(42°)
b = 12 × 0.9563/0.6691
b ≈ 17.15 cm
Verify Your Answer with the Law of Sines CalculatorExample 2: ASA Case (Two Angles and the Included Side)
Problem: In triangle PQR, angle P = 35°, angle R = 62°, and side q = 8 m (the side between angles P and R). Find side p.
Step 1: Find the missing angle.
Since all angles in a triangle sum to 180°: Q = 180° - 35° - 62° = 83°
Step 2: Identify the opposite pairs.
- Side q is opposite angle Q (83°)
- Side p is opposite angle P (35°)
Step 3: Set up and solve.
p/sin(35°) = 8/sin(83°)
p = 8 × sin(35°)/sin(83°)
p = 8 × 0.5736/0.9925
p ≈ 4.62 m
In ASA problems, always find the third angle first! Once you have all three angles, the problem becomes straightforward because you can create any opposite pair you need.
Example 3: The Ambiguous SSA Case
This is where things get interesting—and where many students make mistakes.
Problem: In triangle ABC, side a = 10, side b = 14, and angle A = 30°. Find angle B.
Step 1: Set up the proportion to find angle B.
a/sin(A) = b/sin(B)
10/sin(30°) = 14/sin(B)
Step 2: Solve for sin(B).
sin(B) = 14 × sin(30°)/10
sin(B) = 14 × 0.5/10
sin(B) = 0.7
Step 3: Find angle B.
B = sin⁻¹(0.7) ≈ 44.43°
But wait! Here’s where the SSA case earns its “ambiguous” nickname.
When solving for an angle using the Law of Sines, remember that sine is positive in both the first and second quadrants. This means there could be two possible angles: one acute and one obtuse.
If sin(B) = 0.7, then:
- B₁ ≈ 44.43° (from your calculator)
- B₂ ≈ 180° - 44.43° = 135.57°
Step 4: Check if both solutions are valid.
For B₁ = 44.43°:
- Angle C = 180° - 30° - 44.43° = 105.57° ✓ (valid—all angles positive)
For B₂ = 135.57°:
- Angle C = 180° - 30° - 135.57° = 14.43° ✓ (valid—all angles positive)
Both triangles exist! This problem has two valid solutions.
Understanding the Ambiguous Case Outcomes
The SSA case can produce three different outcomes. Here’s how to determine which applies:
Scenario Analysis for SSA
When you’re given sides a and b, with angle A opposite side a:
No Solution: If a < b × sin(A), no triangle can be formed. The side opposite the known angle is too short to reach the base.
One Solution:
- If a ≥ b (the side opposite the known angle is longer than or equal to the other known side)
- Or if the second possible angle would make the angle sum exceed 180°
Two Solutions: If a < b and both calculated angles produce valid triangles (all angles positive and sum to 180°).
After finding your first angle:
- Calculate the supplement (180° - angle)
- Add the supplement to your given angle
- If this sum is less than 180°, you have two solutions!
Common Mistakes to Avoid
Over years of teaching, I’ve seen students make the same errors repeatedly. Here’s how to avoid them:
Mistake 1: Mixing Up Opposite Pairs
The Law of Sines only works when you pair a side with its opposite angle. Side a must go with angle A, not angle B or C.
Fix: Before setting up your equation, draw the triangle and clearly label which angle is opposite which side.
Mistake 2: Calculator in Wrong Mode
If your answer seems wildly off, check if your calculator is in degree mode or radian mode!
Fix: For most high school problems, ensure your calculator is set to degrees. Check out our guide on degrees vs radians if you need a refresher.
Mistake 3: Forgetting to Check for Two Solutions in SSA
This is the most common error. Students find one valid angle and assume they’re done.
Fix: Always ask yourself, “Could there be a second valid triangle?” Calculate 180° minus your answer and check if it creates a valid triangle.
Mistake 4: Not Finding the Third Angle First in ASA
In ASA problems, students sometimes try to use the Law of Sines immediately, but they don’t have a complete opposite pair yet.
Fix: Always compute the third angle using A + B + C = 180° before applying the Law of Sines.
How Law of Sines Connects to Other Concepts
The Law of Sines doesn’t exist in isolation. Understanding its connections will deepen your trigonometry skills:
- Law of Cosines: Use this when you have SAS or SSS. Some problems require using both laws!
- Triangle Area Formulas: The Law of Sines leads to the area formula: Area = (1/2)ab × sin(C)
- Unit Circle: Understanding why sine values repeat helps explain the ambiguous case
- Inverse Trig Functions: Essential for solving for angles
Practice Problems
Try these problems on your own, then check your work with our calculator:
-
AAS: Triangle ABC has A = 50°, B = 70°, and a = 15. Find b.
-
ASA: Triangle DEF has D = 40°, F = 85°, and e = 20. Find d and f.
-
SSA (Ambiguous): Triangle GHI has g = 8, h = 10, and G = 35°. How many triangles are possible? Find all solutions.
Real-World Applications
The Law of Sines isn’t just for textbook problems. It’s used extensively in:
- Surveying: Calculating distances that can’t be measured directly
- Navigation: Determining positions using triangulation
- Architecture: Designing non-rectangular structures
- Astronomy: Measuring distances to celestial objects
If you’re interested in practical applications, check out our article on surveying with trigonometry.
Quick Reference Summary
| Situation | Use Law of Sines? | Notes |
|---|---|---|
| AAS | ✅ Yes | Find third angle first if needed |
| ASA | ✅ Yes | Always find third angle first |
| SSA | ✅ Yes (with caution) | Check for 0, 1, or 2 solutions |
| SAS | ❌ No | Use Law of Cosines |
| SSS | ❌ No | Use Law of Cosines |
The Law of Sines might seem tricky at first, especially the ambiguous case. But with practice, you’ll develop an intuition for when to use it and how to check your work. Remember: every expert was once a beginner who kept practicing!
Next Steps
Now that you understand the Law of Sines, put your knowledge into practice:
- Work through the practice problems above
- Try our interactive calculator to verify your answers
- Learn about the Law of Cosines for the cases Law of Sines can’t handle
- Review how to solve right triangles if you need to strengthen your foundation
The Law of Sines opens up a whole new world of triangles beyond the familiar right triangle. With the formula memorized, an understanding of the three cases, and awareness of the ambiguous case pitfall, you’re ready to tackle any oblique triangle problem that comes your way. Happy calculating!
