When you have a triangle where the Law of Sines cannot be applied immediately (like knowing all 3 sides, or 2 sides and a contained angle), you need the Law of Cosines Calculator. This tool is the heavy lifter of trigonometry, allowing you to solve for any side or angle in any triangle, regardless of its shape.
Features
Solves SSS and SAS: The two classic cases where Law of Cosines is mandatory.
Equation Breakdown: Displays the specific values substituted into the $c^2 = a^2 + b^2 – 2ab\cos(C)$ formula.
Angle Finding: In SSS mode, calculates angles by rearranging the formula to solve for the cosine term.
Robust Validation: Checks for Impossible Triangles (Triangle Inequality) before computing.
How to Use This Calculator
Choose Case: Select SAS if you have an included angle, or SSS if you only have sides.
Input: Key in the values. For Angle C in SAS, ensure it is the angle *between* sides a and b.
Calculate: Get precise lengths and angles instantly.
Learn: Read the steps to understand the algebraic manipulation.
Formulas
The calculator uses the following mathematical principles:
Standard Form (Finding Side): $$ c^2 = a^2 + b^2 – 2ab \cos(C) $$
Alternate Form (Finding Angle): $$ \cos(C) = \frac{a^2 + b^2 – c^2}{2ab} $$
Area: $$ Area = \frac{1}{2}ab \sin(C) $$
Practical Applications
Robotics: Inverse kinematics (calculating joint angles to reach a coordinate).
Surveying: Determining the distance between two landmarks by measuring distance to each from a third point.
Structural Engineering: Analyzing forces in truss members types.
Tips for Success
Generalized Pythagoras: Notice that if Angle C is 90°, cos(90)=0, and the formula becomes $c^2=a^2+b^2$.
SSS Order: In SSS mode, it calculates the largest angle first using the longest side to avoid ambiguity involved with Law of Sines later.
Degrees: Input angles are always in degrees.
Frequently Asked Questions (FAQ)
Why use Law of Cosines over Sines?
Law of Cosines is necessary when you don’t have a matching Side/Angle pair property to start with (i.e., SAS and SSS cases).
Is the formula different for finding side a or b?
The structure is identical. $$ a^2 = b^2 + c^2 – 2bc \cos(A) $$
Final Words
For the toughest triangle problems, the Law of Cosines Calculator is the definitive solution, bridging the gap between simple geometry and complex trigonometry.
