Trigonometry 4.5 – Trig Proof 1

Related tutorials

• Trigonometry 2.1 — Deriving the Unit Circle → worth a recap before this lesson
• The Line 1 — JC Revision → worth a recap before this lesson
• Trigonometry 4.6 — Trig Proof 2: Prove the Sine Rule `a/sin A = b/sin B = c/sin C` (canonical LCHL proof; 🚨 LOAD-BEARING you must know BOTH the acute-triangle case AND the obtuse-triangle case — exam can ask for either; both methods drop a perpendicular height `h` from a vertex to the opposite side and use SOHCAHTOA to express `h` two ways then equate; the obtuse case additionally uses the **supplementary-angle identity `sin(180° − B) = sin B`** which Paul derives on the fly from unit-circle examples) → the next step after this one
• Trigonometry 4.7 — Trig Proof 3: Prove the Cosine Rule `a² = b² + c² − 2bc cos A` (canonical LCHL proof; 🚨 LOAD-BEARING uses a CIRCLE OF RADIUS c (not the unit circle) centred at vertex A — a point on its edge at angle A has coordinates `(c cos A, c sin A)`; combined with vertex B at `(b, 0)` on the x-axis, the distance from B to that point is side `a`; apply the distance formula (page 18), square, expand, and 🚨 use the Pythagorean identity `cos² A + sin² A = 1` from Trig 4.5 to close — this is the proof where Trig 4.5 pays off) → the next step after this one