Law of Sines Calculator
The Law of Sines is a fundamental trigonometric principle used to solve oblique triangles—triangles that don't contain a right angle. This comprehensive calculator helps students, engineers, surveyors, and mathematics enthusiasts solve triangles using the Law of Sines, handling all cases including the ambiguous SSA case, with step-by-step solutions and properly formatted mathematical formulas.
Triangle Input Values
Enter known values (leave unknown values empty). You need at least one side and its opposite angle.
Angles (in degrees)
Sides
Understanding the Law of Sines
The Law of Sines establishes a relationship between the sides of a triangle and the sines of their opposite angles. This powerful theorem states that the ratio of any side length to the sine of its opposite angle is constant for all three sides of a triangle, providing a method to solve for unknown angles and sides when specific combinations of measurements are known.
Law of Sines Formula
Law of Sines Equation:
Standard Form:
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]
Where:
- \(a, b, c\) = lengths of the sides
- \(A, B, C\) = angles opposite to sides \(a, b, c\) respectively
Alternative Form:
\[\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\]
Both forms are equivalent and useful depending on what you're solving for.
To Find an Unknown Angle:
\[\sin B = \frac{b \sin A}{a}\]
\[B = \arcsin\left(\frac{b \sin A}{a}\right)\]
To Find an Unknown Side:
\[b = \frac{a \sin B}{\sin A}\]
Comprehensive Law of Sines Example
Example: Solve Triangle with AAS Configuration
Given: Angle A = 40°, Angle B = 60°, Side a = 10 units
Find: Angle C, Side b, Side c
Solution:
Step 1: Find Angle C
Using angle sum property of triangles:
\[C = 180° - A - B = 180° - 40° - 60° = 80°\]
Step 2: Find Side b using Law of Sines
\[\frac{a}{\sin A} = \frac{b}{\sin B}\]
\[\frac{10}{\sin 40°} = \frac{b}{\sin 60°}\]
\[b = \frac{10 \times \sin 60°}{\sin 40°} = \frac{10 \times 0.8660}{0.6428} = 13.47 \text{ units}\]
Step 3: Find Side c using Law of Sines
\[\frac{a}{\sin A} = \frac{c}{\sin C}\]
\[\frac{10}{\sin 40°} = \frac{c}{\sin 80°}\]
\[c = \frac{10 \times \sin 80°}{\sin 40°} = \frac{10 \times 0.9848}{0.6428} = 15.32 \text{ units}\]
Answer: C = 80°, b ≈ 13.47 units, c ≈ 15.32 units
When to Use the Law of Sines
The Law of Sines is applicable for specific triangle configurations where you have certain combinations of known angles and sides. Understanding when to use this law versus other methods (like Law of Cosines) is crucial for efficient problem-solving.
| Case | Known Information | Use Law of Sines? | Notes |
|---|---|---|---|
| AAS | Two angles, one side (not between angles) | ✓ Yes | One unique solution, straightforward |
| ASA | Two angles, side between them | ✓ Yes | One unique solution, find third angle first |
| SSA | Two sides, angle opposite one side | ✓ Yes (Caution) | Ambiguous case: 0, 1, or 2 solutions possible |
| SAS | Two sides, angle between them | ✗ No | Use Law of Cosines instead |
| SSS | All three sides | ✗ No | Use Law of Cosines instead |
| AAA | All three angles | ✗ No | Infinite similar triangles, cannot determine size |
Quick Decision Guide: Use Law of Sines when you know at least one side-angle pair (a side and its opposite angle) plus one additional piece of information. If you know two sides and the included angle (SAS) or three sides (SSS), use Law of Cosines instead. For right triangles, basic trigonometry (SOH-CAH-TOA) may be simpler than Law of Sines.
The Ambiguous Case (SSA)
The SSA (Side-Side-Angle) configuration is called the ambiguous case because it can result in zero, one, or two valid triangles depending on the relative lengths of the sides and the measure of the given angle. Understanding this case is essential for correctly solving SSA problems and avoiding missing solutions.
Possible Outcomes in SSA Case
SSA Ambiguous Case Analysis:
Given: Sides \(a\) and \(b\), and angle \(A\) (opposite side \(a\))
Calculate the height: \(h = b \sin A\)
Possible Scenarios:
1. No Solution (if \(A\) is acute):
- When \(a < h\): The side \(a\) is too short to reach the base, no triangle possible
2. One Solution:
- When \(A\) is obtuse or right (≥ 90°) and \(a > b\): Only one triangle possible
- When \(A\) is acute and \(a = h\): Right triangle solution
- When \(A\) is acute and \(a ≥ b\): Only one triangle possible
3. Two Solutions (ambiguous case):
- When \(A\) is acute, \(h < a < b\): Two different triangles satisfy the conditions
- Calculate both: \(B_1 = \arcsin\left(\frac{b \sin A}{a}\right)\) and \(B_2 = 180° - B_1\)
- Verify both create valid triangles (all angles positive, sum = 180°)
Example: Ambiguous SSA Case with Two Solutions
Given: Side a = 7, Side b = 10, Angle A = 30°
Step 1: Calculate height
\[h = b \sin A = 10 \times \sin 30° = 10 \times 0.5 = 5\]
Step 2: Check conditions
Since \(h = 5 < a = 7 < b = 10\) and angle A is acute → Two solutions exist
Step 3: Find angle B (first solution)
\[\sin B = \frac{b \sin A}{a} = \frac{10 \times 0.5}{7} = 0.7143\]
\[B_1 = \arcsin(0.7143) = 45.58°\]
\[C_1 = 180° - 30° - 45.58° = 104.42°\]
\[c_1 = \frac{a \sin C_1}{\sin A} = \frac{7 \times \sin 104.42°}{\sin 30°} = \frac{7 \times 0.9659}{0.5} = 13.52\]
Step 4: Find second solution
\[B_2 = 180° - 45.58° = 134.42°\]
\[C_2 = 180° - 30° - 134.42° = 15.58°\]
\[c_2 = \frac{a \sin C_2}{\sin A} = \frac{7 \times \sin 15.58°}{\sin 30°} = \frac{7 \times 0.2685}{0.5} = 3.76\]
Both triangles are valid!
Triangle 1: A = 30°, B₁ = 45.58°, C₁ = 104.42°, a = 7, b = 10, c₁ = 13.52
Triangle 2: A = 30°, B₂ = 134.42°, C₂ = 15.58°, a = 7, b = 10, c₂ = 3.76
Solving Different Triangle Cases
Different starting configurations require slightly different approaches when applying the Law of Sines. Understanding the specific steps for each case ensures accurate and efficient solutions.
Case 1: AAS or ASA (Two Angles and One Side)
Solution Procedure for AAS/ASA:
- Find the third angle: Use \(C = 180° - A - B\) (angles in triangle sum to 180°)
- Apply Law of Sines: Use the given side and its opposite angle to find other sides
- Calculate remaining sides: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
Note: This case always produces exactly one unique solution.
Case 2: SSA (Two Sides and Non-Included Angle)
Solution Procedure for SSA:
- Calculate height: \(h = b \sin A\) where A is the given angle
- Analyze possibilities: Compare side lengths and height to determine number of solutions
- Find opposite angle: \(\sin B = \frac{b \sin A}{a}\)
- Check for second solution: Calculate \(B_2 = 180° - B_1\) and verify validity
- Complete both triangles: Find remaining angles and sides for all valid solutions
Note: This is the ambiguous case—may have 0, 1, or 2 solutions.
Law of Sines vs. Law of Cosines
Both laws help solve oblique triangles but apply to different situations. Knowing when to use each law is fundamental to efficient trigonometric problem-solving.
| Aspect | Law of Sines | Law of Cosines |
|---|---|---|
| Formula | \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\) | \(c^2 = a^2 + b^2 - 2ab\cos C\) |
| Best For | AAS, ASA, SSA cases | SAS, SSS cases |
| Relates | Sides to opposite angles via sine | All three sides to one angle via cosine |
| Ambiguity | SSA case can have 0, 1, or 2 solutions | Always one unique solution |
| When to Use | When you know angle-side pairs | When you know sides and included angle, or all sides |
| Calculation | Simpler arithmetic, uses ratios | More complex, requires squaring |
Choosing the Right Law: If you can use Law of Sines (have angle-side pair), it's usually simpler. However, for SAS and SSS cases, you must use Law of Cosines. After finding one angle with Law of Cosines, you can switch to Law of Sines to find remaining parts more easily. Some problems may require both laws in sequence.
Real-World Applications
The Law of Sines extends beyond classroom exercises into practical applications across engineering, navigation, surveying, and physics. Understanding these applications contextualizes the mathematical theory.
Practical Uses of Law of Sines
- Navigation and Marine Operations: Calculating distances and bearings between ships, determining positions using triangulation from known landmarks, and planning optimal routes considering angles and distances
- Surveying and Land Measurement: Determining inaccessible distances across rivers or ravines, measuring property boundaries when obstacles prevent direct measurement, and creating accurate topographic maps
- Astronomy: Calculating distances to stars and planets using parallax angles, determining orbital parameters, and measuring celestial object positions using triangulation from different observation points
- Architecture and Construction: Calculating roof angles and rafter lengths for non-standard designs, determining bridge cable tensions from support angles, and planning structural supports for irregular building shapes
- Aviation: Determining aircraft position using radio beacons, calculating wind correction angles for navigation, and planning flight paths considering various approach angles
- Physics and Engineering: Analyzing force vectors in non-perpendicular directions, calculating component forces in mechanical systems, and determining resultant forces from multiple angular inputs
Common Mistakes and How to Avoid Them
- Using Law of Sines for SAS or SSS: These cases require Law of Cosines. Law of Sines cannot directly solve triangles when you know two sides and the included angle, or three sides without any angles.
- Missing the Second Solution in SSA: Always check if the SSA case produces two valid triangles. Calculate the supplementary angle \(180° - B\) and verify it creates a valid triangle before concluding only one solution exists.
- Degrees vs. Radians: Ensure your calculator is in the correct mode. Most problems use degrees, but calculus and advanced mathematics use radians. Mixing modes produces completely wrong answers.
- Inverse Sine Domain Error: If \(\sin B = \frac{b \sin A}{a} > 1\), no triangle exists. The sine function only outputs values between -1 and 1. A result greater than 1 indicates impossible triangle geometry.
- Not Verifying Angle Sum: Always check that \(A + B + C = 180°\). If not, calculation errors occurred or the triangle is impossible with given values.
- Incorrect Angle-Side Pairing: Ensure you're using the side opposite to the given angle. Side \(a\) must be opposite angle \(A\), side \(b\) opposite angle \(B\), etc. Mixing up pairings produces incorrect results.
- Assuming Angles are Acute: Triangles can have obtuse angles (> 90°). When using inverse sine, consider whether the angle should be acute or obtuse based on triangle geometry.
Derivation of the Law of Sines
Understanding where the Law of Sines comes from deepens comprehension and helps remember the formula. The derivation uses the relationship between triangle area and trigonometry.
Law of Sines Derivation:
Method 1: Using Triangle Area
The area of a triangle can be expressed using two sides and the included angle:
\[\text{Area} = \frac{1}{2}ab\sin C = \frac{1}{2}bc\sin A = \frac{1}{2}ac\sin B\]
Since all equal the same area, set them equal:
\[\frac{1}{2}ab\sin C = \frac{1}{2}bc\sin A\]
Simplify: \(ab\sin C = bc\sin A\)
Divide both sides by \(abc\): \(\frac{\sin C}{c} = \frac{\sin A}{a}\)
Similarly, \(\frac{\sin B}{b} = \frac{\sin A}{a}\)
Therefore: \(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\)
Or equivalently: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
Method 2: Using Altitude
Draw altitude \(h\) from vertex to opposite side. From right triangle formed:
\(h = b\sin A = a\sin B\)
Therefore: \(\frac{a}{\sin A} = \frac{b}{\sin B}\)
Repeat for other altitudes to complete the law.
Extended Applications and Special Cases
Using Law of Sines with Right Triangles
While basic trigonometry (SOH-CAH-TOA) typically handles right triangles more simply, the Law of Sines still applies and can provide alternative solution paths.
Right Triangle Special Case: In a right triangle with right angle at C (C = 90°), \(\sin C = \sin 90° = 1\). The Law of Sines becomes: \(\frac{a}{\sin A} = \frac{b}{\sin B} = c\). This means the hypotenuse \(c\) equals \(\frac{a}{\sin A}\) or \(\frac{b}{\sin B}\), which aligns with basic trigonometry: \(a = c\sin A\) and \(b = c\sin B\).
Circumradius and the Law of Sines
The Law of Sines connects to the circumradius (radius of the circle passing through all three vertices) of a triangle, providing geometric insight.
Extended Law of Sines:
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\]
Where \(R\) is the circumradius of the triangle
This means: \(R = \frac{a}{2\sin A} = \frac{b}{2\sin B} = \frac{c}{2\sin C}\)
This relationship is useful in geometry problems involving circles and triangles.
Frequently Asked Questions
What is the Law of Sines?
The Law of Sines states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant for all three sides. The formula is: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), where \(a, b, c\) are side lengths and \(A, B, C\) are the opposite angles. This law is used to solve oblique (non-right) triangles when you know certain angle-side combinations, specifically AAS, ASA, or SSA configurations.
When should you use the Law of Sines?
Use the Law of Sines when you have: 1) Two angles and one side (AAS or ASA case) - produces one unique solution, 2) Two sides and an angle opposite one of them (SSA case) - the ambiguous case that may have 0, 1, or 2 solutions. Do not use Law of Sines for SAS (Side-Angle-Side) or SSS (Side-Side-Side) cases—use Law of Cosines instead. Law of Sines works for all triangle types including acute, obtuse, and right triangles.
What is the ambiguous case in Law of Sines?
The ambiguous case (SSA - Side-Side-Angle) occurs when you know two sides and an angle opposite one of them. This configuration can result in zero, one, or two valid triangles depending on the side lengths and angle. Two solutions exist when: the given angle is acute, the side opposite the angle is shorter than the other given side, and the side opposite is longer than the altitude (\(h = b\sin A\)). Calculate both possible angles: \(B_1 = \arcsin\left(\frac{b\sin A}{a}\right)\) and \(B_2 = 180° - B_1\), then verify each creates a valid triangle.
How do you solve for a missing angle using Law of Sines?
To find a missing angle: 1) Set up the proportion using a known angle-side pair and the side opposite the unknown angle: \(\frac{\sin A}{a} = \frac{\sin B}{b}\), 2) Rearrange to \(\sin B = \frac{b \sin A}{a}\), 3) Use inverse sine: \(B = \arcsin\left(\frac{b \sin A}{a}\right)\), 4) Check if \(\sin B \leq 1\) (if > 1, no solution exists), 5) For SSA case, also check supplementary angle: \(B' = 180° - B\), and verify both solutions create valid triangles where all angles are positive and sum to 180°.
What is the difference between Law of Sines and Law of Cosines?
Law of Sines relates sides to opposite angles using sine ratios: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). It's best for AAS, ASA, and SSA cases. Law of Cosines relates all three sides and one angle: \(c^2 = a^2 + b^2 - 2ab\cos C\). It's best for SAS and SSS cases. Use Law of Sines when you know angle-side pairs; use Law of Cosines when you know two sides and included angle, or all three sides. Law of Cosines always gives one unique solution, while Law of Sines SSA case can be ambiguous.
Why does SSA sometimes have two solutions?
SSA can produce two triangles because when you know two sides and a non-included angle, the second side can "swing" to two different positions while maintaining the same angle and side lengths. Imagine side \(b\) as a compass arc—it may intersect the baseline at two points, creating two different triangles with the same SSA measurements. This only occurs when specific conditions are met: acute given angle, side opposite shorter than other side, but longer than altitude. Both triangles are mathematically valid, though in real-world applications, context determines which solution is physically meaningful.
Can you use Law of Sines for right triangles?
Yes, Law of Sines applies to right triangles, though basic trigonometry (sin, cos, tan) is usually simpler. For a right triangle with right angle C = 90°, since \(\sin 90° = 1\), the Law of Sines simplifies to: \(\frac{a}{\sin A} = \frac{b}{\sin B} = c\). This means \(a = c\sin A\) and \(b = c\sin B\), which are the standard right triangle sine relationships. While mathematically valid, SOH-CAH-TOA is more direct for right triangles. Use Law of Sines for right triangles when working within broader problems involving multiple triangles.
What if the sine value is greater than 1?
If your calculation produces \(\sin B > 1\), no triangle exists with the given measurements. The sine function has a range of [-1, 1], so any value outside this range is mathematically impossible. This typically occurs in SSA cases when the side opposite the known angle is too short to form a triangle—the "no solution" scenario. For example, if you calculate \(\sin B = 1.3\), the given sides and angle cannot form a valid triangle. Recheck your measurements or recognize the configuration is geometrically impossible.
