Law of sines is one of 3 triangles formulas in the trigonometry section of this library, and it is used at high school level.
Why law of sines works
Drop a perpendicular from one vertex to the opposite side. That single height can be written two ways, as b sin A and as a sin B, so the two expressions are equal. Rearranging gives a/sin A = b/sin B, and repeating with a different vertex brings the third pair into the same chain of equal ratios.
What each symbol means
Sides $a,b,c$ lie opposite angles $A,B,C$.
Law of sines: when it holds
An SSA setup may have zero, one, or two valid triangles; angles sum to $180^\circ$.
When it stops applying
The SSA setup, two sides and an angle not between them, is not enough to fix one triangle. Depending on the numbers you can get two triangles, exactly one, or none at all, and the formula happily returns a value in the ambiguous case without warning you.
Law of sines: a worked example
If $A=30^\circ$, $a=5$, $B=45^\circ$, then $b=5\sin45^\circ/\sin30^\circ=5\sqrt2$.
The mistake to avoid
What people do: Students take whatever angle the inverse sine key returns and stop there.
Why it goes wrong: The inverse sine only ever reports angles from −90 to 90 degrees, so it hands back the acute option every time. An obtuse triangle angle simply cannot come out of that key.
Do this instead: After finding one angle, also test its supplement. With a = 5, b = 7, and A = 30 degrees, sin B = 0.7, so B is 44.43 or 135.57 degrees. Both leave a positive third angle, 105.57 and 14.43 degrees, so this problem has two real triangles.
Law of sines: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. Sides $a,b,c$ lie opposite angles $A,B,C$.
- Check the conditions before substituting. An SSA setup may have zero, one, or two valid triangles; angles sum to $180^\circ$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most trigonometry slips.
Where this formula fits
- Subject
- Trigonometry formulas — 18 entries in this library
- Topic
- Triangles
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where law of sines comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Sine and Cosine Rules — the lesson behind this formula: solve non-right triangles from known side and angle data.
- Trigonometry Calculator — check your substitution and the value it produces.
- Study trigonometry — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 18 trigonometry formulas — the full grouped reference, or the complete formula library.
Questions about law of sines
When should I use this instead of the law of cosines?
Use it when your known side and known angle sit across from each other, which happens with ASA, AAS, and SSA. If you have three sides or an angle wedged between two sides, the law of cosines is the one that works.
How do I know whether the second triangle is real?
Subtract both candidate angles from the total of 180 degrees along with the angle you already had. If the leftover third angle is positive, that triangle exists; if it is zero or negative, throw the candidate away.
Can I flip the ratios upside down?
Yes. sin A/a = sin B/b is the same statement, and it is easier to work with when you are solving for an angle, since the unknown ends up on top instead of buried in a denominator.
Does this work in a right triangle too?
It does. With C = 90 degrees, sin C is 1 and the chain gives c = a/sin A, which is just the definition of sine rearranged. The simpler right-triangle ratios are usually faster there.