Law of Sines and Cosines Calculator
Solve any triangle using the Law of Sines and Law of Cosines with our easy-to-use calculator. Find missing sides and angles by entering known values. This tool helps students, teachers, and professionals quickly solve triangle problems.
Law of Sines Calculator
The Law of Sines relates the lengths of sides to the sines of their opposite angles:
a / sin(A) = b / sin(B) = c / sin(C)
Law of Cosines Calculator
The Law of Cosines relates the lengths of sides to the cosine of one angle:
c² = a² + b² - 2ab cos(C)
Combined Calculator
Use both laws together to solve any triangle. Enter any three values (sides or angles).
Triangle Solving Tips
- You need at least 3 known values (sides or angles)
- For Law of Sines, you need at least one side-angle pair
- For Law of Cosines, you need either:
- Three sides (SSS), or
- Two sides and the included angle (SAS)
- Angles in a triangle always sum to 180°
Results
| Element | Value |
|---|---|
| Side a | - |
| Side b | - |
| Side c | - |
| Angle A | - |
| Angle B | - |
| Angle C | - |
| Area | - |
| Perimeter | - |
Solution Method
Enter values and click Calculate to see results.
About the Law of Sines and Cosines
The Law of Sines and Law of Cosines are trigonometric relationships that allow you to solve any triangle (find all unknown sides and angles) when you know certain combinations of sides and angles. These laws are essential tools in trigonometry, geometry, navigation, and many applied fields.
Law of Sines
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:
a / sin(A) = b / sin(B) = c / sin(C)
This law is particularly useful when you know:
- Two angles and one side (AAS or ASA)
- Two sides and a non-included angle (SSA) - though this case may have two solutions
Law of Cosines
The Law of Cosines generalizes the Pythagorean theorem to any triangle:
c² = a² + b² - 2ab cos(C)
This law is particularly useful when you know:
- Three sides (SSS)
- Two sides and the included angle (SAS)
When to Use Each Law
Use Law of Sines when:
- You know two angles and one side (AAS or ASA)
- You know two sides and a non-included angle (SSA) - ambiguous case
Use Law of Cosines when:
- You know three sides (SSS)
- You know two sides and the included angle (SAS)
Ambiguous Case (SSA):
When you know two sides and a non-included angle (SSA), there may be two possible solutions, one solution, or no solution. This is called the "ambiguous case" of the Law of Sines.
Example Problems
Example 1: Law of Sines
Given: Angle A = 30°, Angle B = 45°, Side a = 10
Find: Side b, Side c, Angle C
Example 2: Law of Cosines
Given: Side a = 5, Side b = 7, Angle C = 60°
Find: Side c, Angle A, Angle B