Triangle Solver — Sine & Cosine RulesJEE Main

Two sides + included angle → everything — interactive Mathematics simulation for IIT-JEE.

Concept

Two laws solve every triangle. Cosine rule (three sides + one angle): $a^2 = b^2 + c^2 - 2bc\cos A$ — Pythagoras with a correction term. Sine rule (side–angle ratios): $\frac{a}{\sin A} = 2R$ , tying the triangle to its circumcircle. Area comes free: $\tfrac12 bc\sin A$ .

Key formula

a^2 = b^2 + c^2 - 2bc\cos A, \qquad \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R, \qquad \Delta = \tfrac12 bc\sin A = rs

Derivation

Cosine rule: drop an altitude and Pythagorise the two pieces, or expand $|\vec b - \vec c|^2$ .

Sine rule: the side a subtends angle 2A at the circumcentre, so a = 2R sin A.

Given SAS: cosine rule → a, then sine rule → B, angle sum → C. Heron's Δ = √(s(s−a)(s−b)(s−c)) cross-checks; r = Δ/s and R = abc/4Δ complete the picture.

Scenarios to explore

Triangle Solver — Sine & Cosine Rules — Two sides and an angle → the whole triangle, solved live.

Real-world applications

Navigation & triangulation (GPS geometry).
Resultant of two vectors at an angle IS the cosine rule.
Circumradius/inradius identities power JEE geometry chains.

JEE exam tips

cos A = (b²+c²−a²)/2bc — the rearranged form answers 'find the angle' directly.
Δ = abc/4R and Δ = rs — memorise both; ratio questions use them together.
Projection formula: a = b·cosC + c·cosB — quick and underused.

Common mistakes

Cosine-rule sign: −2bc·cosA (positive when A obtuse makes a² bigger — check!).
Sine-rule ambiguity: asin gives acute; the obtuse supplement may also be valid (SSA case).
Area with the NON-included angle.

Exam traps to avoid

Largest side faces largest angle — sanity check every answer.
In SSA (two sides + non-included angle) TWO triangles may exist — the ambiguous case.