Geometry and Trigonometry Applications
Geometry and Trigonometry Applications
This domain tests the use of geometry and trigonometry to solve practical two- and three-dimensional problems: roofs, frames, mechanisms, circular features, surveying and coordinate layouts. The marks are for correct selection and clear application, not for formula recall alone.
Core rules
- Right triangles: Pythagoras and SOHCAHTOA. Non-right triangles: sine rule a/sin A = b/sin B = c/sin C; cosine rule a² = b² + c² − 2bc cos A for the side opposite the included angle.
- Area of a triangle = ½ab sin C. For circular work, use chord, arc and tangent relationships and the perpendicular from the centre to a chord.
- Keep units consistent and carry more precision than you report. Round only the final answer.
- Check whether the calculator is in degrees or radians, and test whether an obtuse angle is possible before using the sine rule.
- Draw a labelled diagram with knowns and unknowns; a clear free-body or triangle sketch prevents most errors.
Worked example
A roof truss has two rafters of 5 m and 7 m meeting at an included angle of 60°. Find the span and the angle opposite the 7 m rafter. The span is the third side: c² = 5² + 7² − 2(5)(7)cos 60° = 25 + 49 − 35 = 39, so c = 6.245 m. Use the sine rule for the angle opposite the 7 m rafter: sin B/7 = sin 60°/6.245, so sin B = 7 × 0.8660/6.245 = 0.9707. The calculator gives B = 76.1°; the supplementary 103.9° is rejected because the known 5 m side would then require a third angle of 16.1° and a side near 2.0 m, not 5 m.
Traps that cost marks
Using the sine rule when the included angle is known, forgetting calculator mode, rounding before the final answer, ignoring the obtuse-angle check, and assuming a triangle is right-angled.