Cosine rule Calculator
Relate sides and included angle in any triangle.
Formula first
Overview
The Cosine Rule is an extension of the Pythagorean theorem that relates the lengths of the sides of any triangle to the cosine of one of its angles. It allows for the calculation of unknown side lengths or angles in non-right-angled triangles using the Side-Angle-Side (SAS) or Side-Side-Side (SSS) configurations.
Symbols
Variables
a = Side a, b = Side b, C = Angle C, c = Side c
Apply it well
When To Use
When to use: Use this rule when you know two sides and the angle between them (SAS) to find the missing third side. It is also the primary method to find any angle of a triangle when all three side lengths (SSS) are known. This rule is essential when the Sine Rule cannot be applied because no side-angle pair is complete.
Why it matters: This formula is foundational for navigation and surveying, allowing for the calculation of distances between points that are not directly reachable. It is also used in computer graphics and robotics to determine the necessary joint angles for limb positioning or camera orientation in 3D space.
Avoid these traps
Common Mistakes
- Order of operations in a²+b²-2ab...
- Sign errors with cos(obtuse).
One free problem
Practice Problem
A triangle has two sides of length 5 cm and 8 cm. If the angle between these two sides is 60°, calculate the length of the third side.
Solve for:
Hint: Plug the sides into the formula and remember that the cosine of 60 degrees is 0.5.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Wikipedia: Law of cosines
- Britannica: Law of Cosines
- Stewart, J. (2012). Calculus: Early Transcendentals. Cengage Learning.
- Wikipedia: Euclidean geometry
- OCR A-Level Mathematics — Pure (Trigonometry)