\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}Variables
- a, b, c: side lengths
- A, B, C: opposite angles
How to use this formula
Relates each side of a triangle to the sine of its opposite angle.
Important notes
- The ambiguous SSA case can produce zero, one, or two triangles.
- Angles must use a consistent unit.
Quick example
If A = 30°, a = 5, and B = 45°, then b = 5 sin45° / sin30°.
Applicability, worked calculation, and verification
Domain and applicability
Apply Law of Sines only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- The ambiguous SSA case can produce zero, one, or two triangles.
- For the Law of Sines, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- The angle unit and the domain or branch of any inverse trigonometric function must be explicit.
- Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula.
Do not use this formula when
- Do not use Law of Sines when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units.
Boundary and special cases
- For Law of Sines, check zero, negative, and extreme input values before relying on the result.
- When using Law of Sines, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
With A=30°, a=5, and B=45°, b=5sin45°/sin30°=5√2≈7.071.
- Write b/sin45°=5/sin30°.
- Solve for b by multiplying by sin45°.
- Use sin45°=√2/2 and sin30°=1/2.
Independent verification
The larger angle B=45° is opposite the larger side b≈7.071, which is consistent.
Common mistakes
- When copying Law of Sines, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- For the Law of Sines, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step.
Continue the workflow
Use Law of Sines in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Law of Sines.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Passed the core formula indexing gate.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
Frequently asked questions
What is the Law of Sines used for?
Relates each side of a triangle to the sine of its opposite angle.
Can I copy this formula as LaTeX?
Yes. Copy \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.