Sine and Cosine Rule Question Generator
This site's Pythagoras and trigonometry generators both say plainly, in their own limitations, that the cosine rule case — a triangle with no right angle — is not covered. This is that case. Three question shapes: two sides and the angle between them, find the third side (the cosine rule); all three sides, find one angle (the cosine rule rearranged); or two angles and the side between them, find another side (the sine rule). SOHCAHTOA does not apply to any of these, because none of the triangles has a right angle to anchor it.
What this generator does
Draws whole-number side lengths and whole-degree angles directly (redrawing if the resulting triangle would be a near-degenerate sliver), computes the textbook answer from the cosine rule or sine rule, and then places the same triangle as coordinates to measure the target side or angle a second, independent way — a side by the plain distance formula, an angle by the dot-product formula rather than the cosine rule restated. A question is only returned once the rule's answer and the coordinate measurement agree to within a tight tolerance.
How to use this tool
- Choose the cosine rule for a side or an angle, or the sine rule for a side, then a level and a count.
- Work out the missing side or angle using the cosine rule or the sine rule, as the question calls for.
- Reveal the answers to check your own, or the working to see the rule applied.
- Seed the set if two people should get the identical questions.
Understanding the controls
- Question type
- Two sides plus the angle between them (cosine rule, find a side); all three sides (cosine rule, find an angle); or two angles plus the side between them (sine rule, find a side).
- Level
- Gentle keeps sides shorter and angles away from the extremes; standard and stretch widen both, allowing narrower and wider triangles.
- How many questions
- Between 1 and 20 — ten is a reasonable page once the working is shown for each one.
- Seed (optional)
- Any word reproduces the identical set of questions, which is how you give two people the same sheet.
Common use cases
- Practice at the cosine rule, both for a missing side and for a missing angle
- Practice at the sine rule for a missing side
- The non-right-angled companion to this site's Pythagoras and SOHCAHTOA generators
- A worksheet that never runs out of fresh, well-formed triangles
- Giving a class the identical triangles, so a worked diagram on the board matches every printed sheet
How this generator works
The given numbers in each question are drawn directly — whole side lengths, whole-degree angles — rather than built backwards from a chosen answer, so a triangle is rejected and redrawn whenever it would be a near-degenerate sliver: any angle under eight degrees, or three sides that satisfy the triangle inequality by only a hair. Given values in hand, the taught rule computes an answer — the cosine rule for a side or an angle, the sine rule for a side. That answer is then checked against a second, unrelated computation: the same triangle is placed as coordinates (an SSS triangle via a circle-intersection construction, an SAS triangle by placing the included angle directly, an ASA triangle by intersecting the two rays from either end of the known side), and the target quantity is measured from those coordinates — a side with the ordinary distance formula, an angle with the dot-product/arccos formula, neither of which is the rule formula rearranged. A question is kept only when the two answers agree to within a tight numerical tolerance.
Randomness and fairness
The given sides and angles come from your browser's cryptographic random source. Giving a seed replaces that source with a repeatable one, so the same set of questions can be reproduced; the seed is for reproducibility, not cryptographic strength.
For how randomness is produced across the whole site, see how Generate Random works.
Limitations and good to know
- Every given side length and angle is a whole number; the answer itself is not, and is shown rounded (two decimal places for a side, one for an angle).
- Triangles within eight degrees of degenerate, in any angle, are rejected rather than offered as an unusually hard case.
- There is no answer box to type into — the worked answer is shown for you to check your own figures against, not graded automatically.
- The ambiguous case of the sine rule (where two different triangles can fit the same two sides and one non-included angle) is not offered — every question here has exactly one triangle that fits the given values.
Privacy and your data
Every triangle, its rule-based answer and its coordinate cross-check are all computed in your browser. Nothing you choose here — question type, level, seed or count — is transmitted or retained.
Related generators
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- Trigonometry Question GeneratorSOHCAHTOA questions that name the two sides involved, including inverse questions that find an angle.
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