Goal and prerequisites
Handle a non-right triangle or a three-dimensional diagonal using the correct geometric relationship. Review right triangles before starting.
Method and its conditions
For two sides and their included angle, use c²=a²+b²−2 ab cos C. Use a/sin A=b/sin B when an opposite side-angle pair is known. Triangle area is ab sin C/2. In a cuboid, find a face diagonal and then a space diagonal.
Worked example: show the decisions
Two sides 4 and 6 enclose 60°. The third side squared is 16+36−48(1/2)=28, so its length is 2√7. If sides 6 and 8 enclose 30°, the area is(1/2)×6×8×(1/2)=12. Neither calculation assumes a right angle.
Contrast: diagnose an incorrect approach
Pythagoras cannot replace the cosine rule unless the included angle is 90°. A three-dimensional diagonal is longer than a face diagonal: a cuboid with sides 2,4,4 has diagonal√(4+16+16)=6. Conversely, if a cuboid has diagonal 15 and known perpendicular edges 5 and 10, its remaining edge is√(15²−5²−10²)=10; the positive radicand confirms a feasible positive edge. In the sine rule, pair each side with its opposite angle rather than the adjacent one.
Transfer: try a changed situation
A triangle has side 5 opposite 30° and side b opposite 45°. Find b², and explain why this is not a right-triangle ratio problem.
Transfer solution and interpretation
The sine rule gives b=5 sin 45°/sin 30°=5√2, hence b²=50. The third angle is 105°, so the triangle is not right angled.
A triangle has sides 3 cm and 6 cm enclosing 60°. Its third side is c cm. Find c².
- Use the cosine rule with the included angle 60°, whose cosine is 1/2.
- Using the stated quantities: 3^2+6^2-2×3×6×(1/2) = 27.
- Check the domain, units and whether the answer is an exact value, bound or estimate.
Compare the other answer choices
Use the squared side difference as though the included angle were 0°.
Apply Pythagoras despite the included angle not being 90°.
Add the cosine term instead of subtracting it.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
0580 syllabus for exams in 2025–2027