Law of Sines and Cosines: Solve Any Triangle with 39 Flashcards (Core vs. Proofs)

The law of sines and the law of cosines are the two tools that let you solve any triangle, not just right ones. The Flip It deck Law of Sines & Cosines (code CQ5924, 39 cards) covers them — and then goes much further, into spherical, hyperbolic and n-dimensional versions. Knowing which cards are the exam core and which are the encyclopedia is the whole trick.

The core: 9 cards that solve triangles

Learn these first (cards 1-7 and 25-27 in the deck). They are the part of the topic that appears on every precalculus and trigonometry exam.

The extension layer

Everything else is proof technique or generalization, drawn from a university-level treatment: Euclid's Elements II.12 and II.13, the French name théorème d'Al-Kashi, proofs by Cartesian coordinates, altitudes (c = a cos β + b cos α), dot products, Ptolemy's theorem, power of a point, Thales's theorem and the inscribed angle theorem; the tangent-secant and chord theorems; area T = abc/(4R) and Heron's formula; the isosceles form c² = 2a²(1 − cos γ); the small-angle form c² = (a − b)² + 2ab(1 − cos γ) that avoids catastrophic cancellation; the law of tangents and Mollweide's formula; the spherical laws of sines and cosines; both hyperbolic laws of cosines; the tetrahedron analogue; the polar-sine law for simplices; and the curvature function Sₖ(x) that unifies all three geometries.

A study plan in two passes

Pass 1 — the solver. Suspend cards 8-24 and 28-39. Run only the nine core cards until the choice between sines and cosines is automatic: two sides and the angle between them, or three sides → cosines; a known angle facing a known side → sines.

Pass 2 — the proofs. Unsuspend the rest if your course asks for derivations or non-Euclidean geometry. Group the proofs by the theorem they borrow (circle theorems 28-33, algebraic proofs 15, 20, 22, 31).

Cards to fix before your first review

• ASA is missing. The deck handles SAS, SSS and AAS, but has no card for two angles and the side between them. Add one: third angle from 180°, then law of sines.
• Card 31 asks for the wrong law. The front says law of sines using dot products, but the back, c² = (a − b)·(a − b), proves the law of cosines. Rename the front.
• The ambiguous case has no count. Card 5 gives the condition c sin α < a < c for two triangles; add the other outcomes: a = c sin α gives one right triangle, a < c sin α gives none.
• Zero worked examples. Not one card turns numbers into an answer. Write three yourself, e.g. a = 7, b = 9, γ = 60° → c² = 49 + 81 − 63 = 67, so c ≈ 8.19.

For the right-triangle groundwork, see SOH-CAH-TOA and the unit circle. For the angle-sum identities used in the proofs, see trigonometric identities.

Find CQ5924 — Law of Sines & Cosines in the library on Flip It.