Law of Sines Calculator
Solve triangles using the sine rule: AAS, ASA, and SSA cases with step-by-step solutions
Triangle Calculator
Triangle Solution
Triangle Properties
Sides
Angles
Additional Properties
Example: AAS Case
Problem
In a triangle, side a = 5, angle α = 30°, and angle β = 60°. Find the remaining sides and angle.
Solution
Step 1: Find angle γ = 180° - 30° - 60° = 90°
Step 2: Apply law of sines:
b = a × sin(β) / sin(α) = 5 × sin(60°) / sin(30°) = 5 × 0.866 / 0.5 = 8.66
c = a × sin(γ) / sin(α) = 5 × sin(90°) / sin(30°) = 5 × 1 / 0.5 = 10
Result: b = 8.66, c = 10, γ = 90° (Right triangle!)
Law of Sines Formula
The ratio of any side to the sine of its opposite angle is constant
Alternative Forms
a = b × sin(α) / sin(β)
sin(α) = a × sin(β) / b
α = arcsin(a × sin(β) / b)
When to Use Law of Sines
SSA Ambiguous Case
- • Angle α is acute (< 90°)
- • Side a < side b
- • a > b × sin(α)
Understanding the Law of Sines
What is the Law of Sines?
The law of sines (also known as the sine rule) 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. This powerful relationship allows us to solve triangles when we know certain combinations of sides and angles.
Applications
- •Navigation and surveying
- •Engineering and architecture
- •Physics problems involving vectors
- •Computer graphics and game development
Law of Sines vs Law of Cosines
Use Law of Sines when you have:
- • Two angles + one side (AAS/ASA)
- • Two sides + one opposite angle (SSA)
Use Law of Cosines when you have:
- • Three sides (SSS)
- • Two sides + included angle (SAS)
Historical Note
The law of sines was known to ancient mathematicians, including Claudius Ptolemy in the 2nd century CE. The modern trigonometric form was developed during the Islamic Golden Age by mathematicians like Al-Biruni and Nasir al-Din al-Tusi.