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Principle deep dive

Sine rule + Cosine rule

a/sin A = 2R and c² = a² + b² − 2ab cos C. The hardest principle in the live bank — 57% of tagged questions are HARD. Drives every "solve the triangle" problem; also surfaces in Height & Distance and inside trig-laden determinants and in-circle problems.

questions in the bank
42
tagged HARD
57%
chapter spread
2
worked examples below
4

When to reach for it

Triangle with some sides and angles known; solve for the rest, or find area / inradius / circumradius.

Why this principle matters

Sine rule: a / sin A = b / sin B = c / sin C = 2R, where R is the circumradius. Cosine rule: c² = a² + b² − 2ab cos C (and cyclic permutations). Between them, every 'solve the triangle' problem in NDA is one or two substitutions.

Which rule to reach for? Sine rule works when you have a side opposite a known angle (SAS, AAS). Cosine rule works when you have all three sides (SSS) or two sides and the included angle (SAS-with-the-angle-between). The 2R version of sine rule is also the bridge to the circumradius — useful for in-radius / circumradius compound questions.

Area: (1/2)·a·b·sin C — derived directly from sine rule. Heron's formula s(s − a)(s − b)(s − c) where s = (a+b+c)/2 is the SSS-only alternative, useful when you don't want to compute an angle first.

4 worked examples from the bank

Each example demonstrates the principle on a real past-year question. Click to reveal the answer, then the solution.

Example 1Properties of TriangleEASY
What is the area of the triangle ABCABC with sides a=10a = 10 cm, c=4c = 4 cm and angle B=30°B = 30°?

[Q34 · Apr · 2021]

Example 2Properties of TriangleMODERATE
The angles A, B and C of a triangle are in the ratio 1:1:4. If the longest side of the triangle is 3 units, then what is the perimeter of the triangle?

[Q36 · Apr · 2026]

Example 3Properties of TriangleMODERATE
In a triangle ABC, if aa, bb and cc are the lengths of the sides opposite to the angles A, B and C respectively, then what is sin⁡(A−B)sin⁡(A+B)\dfrac{\sin(A-B)}{\sin(A+B)} equal to?

[Q37 · Apr · 2026]

Example 4Properties of TriangleHARD
In a triangle ABC, a=4,b=3,c=2a = 4, b = 3, c = 2. What is cos⁡3C\cos 3C equal to?

[Q50 · Sep · 2022]

Variants to recognise

Same principle, different surfaces. Pattern-match these on test day.

  • Sine rule: a / sin A = 2R

    R is the circumradius. Diameter of the circumscribed circle = 2R.

  • Cosine rule: c² = a² + b² − 2ab cos C

    Pythagorean theorem generalised — when C = π/2, cos C = 0 and you recover c² = a² + b².

  • Area = (1/2) · a · b · sin C

    Most common area formula. Variants: (1/2)·base·height (when height is given), or Heron's for SSS.

  • Projection formula: a = b cos C + c cos B

    Often used to eliminate cos terms when the cosine rule produces unpleasant algebra.

Drill every sine rule + cosine rule question

42 questions from the bank — paginated, with cart and Word-export support.

Drill the 42 questions

Related principles

Often combined with this one — drill these next if you found the examples above tractable.