Law of Sines Calculator
Solve any triangle using the Law of Sines with step-by-step solutions and visual representation
What This Calculator Does
This advanced Law of Sines calculator helps you solve triangles when you know:
- ASA (Angle-Side-Angle): Two angles and the side between them
- AAS (Angle-Angle-Side): Two angles and a non-included side
- SSA (Side-Side-Angle): Two sides and a non-included angle (ambiguous case)
The calculator automatically detects the ambiguous case (SSA) and provides all possible solutions with detailed step-by-step explanations.
Enter Your Triangle Values
Triangle Visualization
Solution
Law of Sines Formula
The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides and angles:
Or equivalently:
Where:
- \(a, b, c\) are the lengths of the sides of the triangle
- \(A, B, C\) are the angles opposite to sides \(a, b,\) and \(c\) respectively
Key Properties
- The sum of all angles in a triangle equals \(180°\): \(A + B + C = 180°\)
- Each side is proportional to the sine of its opposite angle
- Can be used to find unknown sides or angles when certain combinations are known
When to Use Law of Sines (vs Law of Cosines)
| Scenario | Use Law of Sines | Use Law of Cosines |
|---|---|---|
| ASA - Two angles and included side | ✓ Yes | ✗ No |
| AAS - Two angles and non-included side | ✓ Yes | ✗ No |
| SSA - Two sides and non-included angle | ✓ Yes (Ambiguous) | ✓ Yes (Alternative) |
| SAS - Two sides and included angle | ✗ No | ✓ Yes |
| SSS - Three sides | ✗ No | ✓ Yes |
Decision Guide
Use Law of Sines when:
- You know two angles and any side (ASA or AAS)
- You know two sides and an angle opposite one of them (SSA - but beware of ambiguous case)
- You need to find an angle when you know its opposite side and another angle-side pair
Use Law of Cosines when:
- You know three sides (SSS)
- You know two sides and the included angle (SAS)
- You want to avoid the ambiguous case in SSA scenarios
Ambiguous Case (SSA) Explained
The SSA (Side-Side-Angle) case is called the "ambiguous case" because it can result in zero, one, or two valid triangles depending on the given measurements.
Why Is It Ambiguous?
When you know two sides and an angle that is NOT between them, the third side can potentially "swing" to create two different triangles, one triangle, or no triangle at all.
The Three Possible Outcomes
Case 1: No Triangle Exists
This occurs when the side opposite the known angle is too short to reach the third vertex.
Condition: If \(a < b \cdot \sin(A)\), no triangle is possible.
Example: \(a = 5\), \(b = 10\), \(A = 30°\)
Check: \(b \cdot \sin(A) = 10 \times \sin(30°) = 10 \times 0.5 = 5\)
Since \(a = 5\) is not greater than \(5\), no valid triangle exists.
Case 2: Exactly One Triangle
This occurs when:
- The known angle \(A \geq 90°\) (obtuse or right), OR
- The side opposite the known angle is longer than or equal to the other known side: \(a \geq b\)
Example: \(a = 10\), \(b = 8\), \(A = 30°\) → One solution since \(a > b\)
Case 3: Two Triangles (Ambiguous)
This occurs when the side opposite the known angle is shorter than the other known side but long enough to form a triangle.
Condition: If \(b \cdot \sin(A) < a < b\) and \(A\) is acute, two solutions exist.
Example: \(a = 7\), \(b = 10\), \(A = 30°\) → Two solutions
The two solutions have angles \(B_1\) and \(B_2\) where \(B_2 = 180° - B_1\)
How This Calculator Handles SSA
Our calculator automatically:
- Detects whether the SSA case will produce 0, 1, or 2 solutions
- Calculates all valid solutions
- Provides warnings when the ambiguous case applies
- Shows step-by-step work for each possible triangle
Worked Examples
Triangle 2: \(B \approx 134.42°\), \(C \approx 15.58°\), \(c \approx 3.76\)
Frequently Asked Questions
The Law of Sines is used to solve triangles when you know either two angles and one side (ASA or AAS) or two sides and a non-included angle (SSA). It's particularly useful for finding unknown angles or sides in oblique (non-right) triangles.
Use the Law of Sines when you know two angles and any side (ASA or AAS), as it provides a more direct solution. Use the Law of Cosines when you know three sides (SSS) or two sides and the included angle (SAS). For SSA cases, either law can work, but Law of Cosines avoids the ambiguous case issue.
The ambiguous case occurs with SSA (Side-Side-Angle) triangles, where the given information can result in zero, one, or two valid triangles. This happens because when you know two sides and an angle that's not between them, there may be multiple ways to complete the triangle, or the triangle may be impossible to form.
Yes, the Law of Sines works for right triangles, but basic trigonometry (SOH-CAH-TOA) is usually simpler and more direct. However, the Law of Sines is primarily designed for oblique triangles where standard right triangle trigonometry doesn't apply.
This calculator accepts angles in degrees, which is the most common unit for triangle problems. Make sure all your angle measurements are in degrees for accurate results. The calculator will also output angles in degrees.
An SSA triangle has no solution when the side opposite the known angle is too short to reach the third vertex. Mathematically, this occurs when a < b × sin(A). The calculator will detect this and inform you that no valid triangle exists with the given measurements.
The calculator provides results rounded to 2 decimal places for practical use. Internally, it uses high-precision floating-point arithmetic. For most educational and practical purposes, this level of precision is more than sufficient.
Yes! This calculator is designed as a learning tool. It shows complete step-by-step solutions so you can understand the process, not just get the answer. Use it to check your work and learn the methodology for solving Law of Sines problems.