Law of Sines Calculator
Calculate missing sides and angles in any triangle using the Law of Sines formula.
Law of Sines Formula:
sin(A)/a = sin(B)/b = sin(C)/c
Where a, b, c are the sides and A, B, C are the angles opposite to them
This will calculate angle B and sides a and c
This will calculate angle C and sides b and c
This will calculate angle B, angle C, and side c
Note: The SSA case may result in 0, 1, or 2 possible triangles (ambiguous case)
Triangle Solution
Ambiguous Case Detected
Your inputs result in two possible triangles (SSA ambiguous case).
Solution 1:
Solution 2:
Triangle Visualization
Note: Visualization is not to scale
Triangle Visualizations (Ambiguous Case)
Solution 1:
Solution 2:
Note: Visualizations are not to scale
Error
Understanding the Law of Sines
The Law of Sines is a fundamental formula in trigonometry that relates the sides of a triangle to the sines of the opposite angles. It's particularly useful for solving triangles when you have certain combinations of sides and angles but not enough information to use more direct methods.
The Formula
The Law of Sines states that the ratio of the sine of an angle to the length of the side opposite that angle is the same for all three angles and sides of a triangle:
sin(A)/a = sin(B)/b = sin(C)/c
Where A, B, and C are the angles of the triangle, and a, b, and c are the sides opposite to those angles, respectively.
When to Use the Law of Sines
The Law of Sines is particularly useful in the following scenarios:
- ASA (Angle-Side-Angle): When you know two angles and the side between them
- AAS (Angle-Angle-Side): When you know two angles and a side that is not between them
- SSA (Side-Side-Angle): When you know two sides and an angle opposite to one of them (the ambiguous case)
The Ambiguous Case (SSA)
The SSA case is called the "ambiguous case" because it can result in zero, one, or two possible triangles:
- No solution: When the given side opposite to the angle is too short, no triangle can be formed
- One solution: When the given side opposite to the angle is exactly the right length, or when the angle is 90°
- Two solutions: When the given side opposite to the angle is of intermediate length, two different triangles can be formed
Understanding the Ambiguous Case:
In the SSA case, if we know side a, side b, and angle A:
- If a > b·sin(A), there is no solution
- If a = b·sin(A), there is exactly one solution (angle B = 90°)
- If b > a > b·sin(A), there are two possible solutions
- If a > b, there is exactly one solution
This calculator handles all of these cases automatically.
Practical Applications
The Law of Sines has numerous practical applications:
- Navigation: Calculating distances and bearings in marine and aviation navigation
- Surveying: Determining distances and angles in land surveying
- Astronomy: Calculating distances between celestial objects
- Physics: Resolving vectors in mechanics and engineering problems
- Architecture: Designing structures with non-right angles
How to Use This Calculator
- Select the calculation type based on what information you have:
- Angle-Side-Angle (ASA): Enter two angles and the side between them
- Angle-Angle-Side (AAS): Enter two angles and a side not between them
- Side-Side-Angle (SSA): Enter two sides and the angle opposite to one of them
- Enter the required values in the appropriate fields
- Click "Calculate Triangle" to see the results
- The calculator will display all remaining sides and angles, along with a visual representation of the triangle
- For the SSA case, if two solutions exist, both will be shown
- Use the "Clear All" button to reset the calculator
Frequently Asked Questions
What's the difference between the Law of Sines and the Law of Cosines?
The Law of Sines relates the sides of a triangle to the sines of the opposite angles, while the Law of Cosines relates the square of a side to the squares of the other two sides and the cosine of the included angle. The Law of Sines is best used when you know ASA, AAS, or SSA, while the Law of Cosines is best used for SAS, SSS, or to find the third angle when you know all three sides.
Why is the SSA case called the "ambiguous case"?
The SSA case is ambiguous because, depending on the given values, it can result in zero, one, or two possible triangles. This happens because you're essentially using the side opposite to the known angle as a "swing arm" that can potentially create two different triangles. This ambiguity doesn't exist in the ASA, AAS, SAS, or SSS cases.
How accurate is this calculator?
This calculator uses standard trigonometric functions and provides results with high precision. However, very small or very large values may lead to rounding errors. If you're working on a critical application requiring extreme precision, always verify results with alternative methods.
Can the Law of Sines be used for right triangles?
Yes, the Law of Sines works for all triangles, including right triangles. In fact, for right triangles, it simplifies to the more familiar trigonometric ratios of sine, cosine, and tangent. However, for right triangles, you might find it simpler to use the Pythagorean theorem and basic trigonometric functions.
What happens if I enter values that don't form a valid triangle?
The calculator checks for various constraints, such as the triangle inequality theorem and that angles sum to 180°. If your inputs don't satisfy these conditions, you'll receive an error message explaining why a valid triangle cannot be formed with those values.