The ambiguous sine-rule case and valid triangle counts
| English | 中文 | Pinyin |
|---|---|---|
| ambiguous case/æmˈbɪɡjuːəs keɪs/ | 歧义情形 | qí yì qíng xíng |
Two sides and a non-included angle can describe two different plots of land. How do we check whether both triangles exist?
- Two sides and a non-included angle can describe two different plots of land. How do we check whether both triangles exist?
- This lesson studies ambiguous case 歧义情形: Given triangle data that can produce two different valid triangles.
Choose the mathematical structure
- Let A be opposite a, with b another known side. The sine rule gives sinB=b sinA/a. If this ratio is above 1 there is no triangle. Otherwise test both B=arcsin(b sinA/a) and its supplement 180°−B. Retain only positive C=180°−A−B; when B=90° the two branches coincide. Every side must be positive.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines ambiguous case?
Given triangle data that can produce two different valid triangles.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Take A=30°, a=7 and b=10. Then sinB=5/7, so B≈45.5847° or 134.4153°. Both give positive C, about 104.4153° or 15.5847°. The corresponding third sides are c=5√3+2√6≈13.5592 and c=5√3−2√6≈3.7613. To derive them, place A at the origin and C=(10cos30°,10sin30°)=(5√3,5); B=(c,0). The condition BC=7 gives (c−5√3)²+25=49. Both intersections lie on the positive baseline. Their areas are bc sinA/2=25√3/2±5√6, so the same data does not determine one area.
The ambiguous sine-rule case and valid triangle counts
Let A be opposite a, with b another known side
Justify each triangle candidate or transformed key point against its defining conditions.
How many triangles fit A=30°, a=7 and b=10?
h=10sin30°=5; 5<7<10 gives two positive baseline intersections.
Test a tempting shortcut
- A second inverse-sine value is only a candidate: it may make the remaining angle zero or negative. For acute A and h=b sinA, the cases are a<h: none; a=h: one right triangle; h<a<b: two; a≥b: one. At a=b the extra baseline intersection is c=0, a degenerate shape. For right or obtuse A, a must exceed b for one triangle; otherwise none.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Two inverse-sine values always give two valid triangles. This claim is false. Explain which definition or assumption it violates.
How many fit A=30°, a=5 and b=10?
a=h=5 gives one right triangle; the inverse-sine branches coincide.
Two inverse-sine values always give two valid triangles.
A second inverse-sine value is only a candidate: it may make the remaining angle zero or negative. For acute A and h=b sinA, the cases are a<h: none; a=h: one right triangle; h<a<b: two; a≥b: one. At a=b the extra baseline intersection is c=0, a degenerate shape. For right or obtuse A, a must exceed b for one triangle; otherwise none.
Interpret a new situation
- In the example h=5: changing a to 5 gives one right triangle, to 4 gives none, and to 10 gives one with A=B=30°. Check the angle sum, side order and cosine-rule reconstruction for every candidate. State both configurations when the given information does not select one, and keep exact lengths/areas where possible.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
How many fit A=30°, a=4 and b=10?
a=4<h=5 makes sinB=5/4>1, so no triangle exists.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · E. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
Given triangle data that can produce two different valid triangles. Choose the relationship, show the method, check its assumptions and interpret the result.