Sine rule Calculator
Relate sides and angles in any triangle.
Formula first
Overview
The Sine Rule, also known as the Law of Sines, defines the constant ratio between the side lengths of a triangle and the sine of their opposite angles. This relationship applies to all triangles, including non-right oblique triangles, making it a more versatile tool than basic SOH-CAH-TOA ratios.
Symbols
Variables
a = Side a, A = Angle A, b = Side b, B = Angle B
Apply it well
When To Use
When to use: Apply this rule when you are given a side and its opposite angle along with at least one other side or angle. It is the primary method for solving Angle-Angle-Side (AAS) and Angle-Side-Angle (ASA) cases, as well as the ambiguous Side-Side-Angle (SSA) case.
Why it matters: This equation is essential for triangulation in land surveying, coastal navigation, and astronomy to determine distances to remote objects. It also allows engineers to calculate stress vectors in non-perpendicular structural frameworks.
Avoid these traps
Common Mistakes
- Using radians incorrectly.
- Inverting one side of the ratio.
One free problem
Practice Problem
In triangle ABC, find the length of side a if angle A is 40°, side b is 10 units, and angle B is 60°.
Solve for:
Hint: Rearrange the formula to solve for side a: a = (b × sin A) / sin B.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Wikipedia: Law of sines
- Britannica: Law of sines
- Neill, H. (2016). Pure Mathematics 1 (Cambridge International AS & A Level Mathematics). Cambridge University Press.
- IUPAC Gold Book, 'dimensionless quantity'
- Law of sines Wikipedia article
- Spherical trigonometry Wikipedia article
- Euclidean geometry Wikipedia article
- Degenerate triangle Wikipedia article