The Law of Cosines
Applied trigonometry extends right-triangle reasoning to oblique triangles, polar coordinates, complex-number form and vectors, creating a unified language for magnitude and direction.
What this article covers
The supplied source develops The Law of Cosines as part of a wider algebra and trigonometry sequence. This handbook article consolidates the section into definitions, rules, formulae, decision methods and verification practices. It deliberately replaces named source examples with neutral technical examples while preserving the mathematical content.
The emphasis is on knowing why a method applies, not just carrying out a sequence of keystrokes. When a numerical result is produced, the final step is interpretation: what does the sign, interval, magnitude, unit, graph feature or domain restriction mean?
Learning outcomes
- SAS cases: Use the law of cosines when two sides and their included angle are known.
- SSS cases: Use it to find an angle when all three side lengths are known.
- Pythagorean connection: When the included angle is 90°, the law reduces to the Pythagorean theorem.
- Largest-angle check: The largest angle lies opposite the largest side, providing a useful reasonableness check.
Core handbook notes
SAS cases
Use the law of cosines when two sides and their included angle are known.
SSS cases
Use it to find an angle when all three side lengths are known.
Pythagorean connection
When the included angle is 90°, the law reduces to the Pythagorean theorem.
Largest-angle check
The largest angle lies opposite the largest side, providing a useful reasonableness check.
Combined workflow
After one angle-side pair is established, the law of sines may finish the triangle efficiently.
Rounding control
Clamp tiny numerical rounding excursions if calculating inverse cosine from values near ±1.
Formula and notation panel
Use these relationships only when their domains and stated conditions are satisfied. Mathematical formulae are general principles; any values used in the worked example are illustrative.
c²=a²+b²-2ab cosCcosC=(a²+b²-c²)/(2ab)Method: a reliable solving workflow
- 1
Identify whether sas cases is the controlling idea in the problem and list the known values, unknowns, units and domain restrictions.
- 2
Translate the information into the notation used for sss cases; keep symbolic structure intact before substituting numbers.
- 3
Apply the relevant rule or formula, showing intermediate algebra so sign changes, excluded values and transformations remain auditable.
- 4
Use largest-angle check to interpret the result graphically or structurally, not merely as an isolated number.
- 5
Verify the result using combined workflow, substitution, an independent calculation, graph behaviour or a dimensional check as appropriate.
Worked example
Problem. Given a=5,b=7,C=60°, find c.
Method and result. c²=25+49-70(0.5)=39, so c=√39≈6.245.
The numbers are illustrative for learning. They are not engineering acceptance criteria, tolerances or standards.
Engineering and technical applications
The source is a general mathematics text. The applications below are neutral engineering-oriented extensions of the same mathematical principles rather than source requirements or standards.
| # | Application area | How to use the mathematics safely |
|---|---|---|
| 1 | force and displacement vectors | Use the mathematics as a model, retain units, state assumptions and verify the result independently where practical. |
| 2 | navigation and directional geometry | Use the mathematics as a model, retain units, state assumptions and verify the result independently where practical. |
| 3 | polar representations | Use the mathematics as a model, retain units, state assumptions and verify the result independently where practical. |
| 4 | oblique-triangle measurement | Use the mathematics as a model, retain units, state assumptions and verify the result independently where practical. |
Decision guide
When several techniques appear possible, prefer the method that exposes structure and preserves exactness. For example, factor before expanding if factorisation reveals zeros; use an exact special-angle value before a decimal approximation; simplify symbolically before substituting repeated numerical values; and state excluded values before cancelling rational factors.
Technology is best used as a verification and exploration tool. A graph can reveal missed roots or unreasonable behaviour, and a calculator can evaluate difficult arithmetic, but neither replaces a clear statement of the model, domain, units and algebraic logic.
Common mistakes and failure modes
- Applying a familiar rule before identifying whether the problem is actually a the law of cosines problem.
- Dropping parentheses or a sign during substitution, expansion, factorisation or rearrangement.
- Ignoring domain restrictions, undefined values, endpoint inclusion or principal-value conventions.
- Rounding too early and then treating a rounded intermediate result as exact.
- Accepting a calculator output without checking algebraic structure, units or plausibility.
A strong technical calculation is auditable. Someone else should be able to follow the variable definitions, reproduce the algebra, identify any approximation and understand why the final answer is admissible.
Verification checklist
Practice prompts
Concept check
Explain the difference between the mathematical object being studied in this article and the nearest related concept from the same learning path. State at least one condition that determines which method is valid.
Symbolic check
Choose one formula from the panel, rearrange it for a different variable where meaningful, and identify every value that would make the rearranged expression undefined or outside the real-number domain.
Graph or structure check
Predict the qualitative behaviour before calculating: signs, intercepts, symmetry, end behaviour, monotonicity, periodicity or feasible region as appropriate to the topic. Then compare with a calculated or plotted result.
Applied check
Create a small engineering example using consistent SI units. Solve it, report the result with sensible precision, and state which assumptions would need confirmation before the calculation could support a real design decision.