Plane Geometry - Bruce E. Shapiro

SECTION 38. THE PYTHAGOREAN THEOREM 195Definition 38.5 (Trigonometry) Let △ABC be a right triangle withright angle at vertex C, and let θ = ∠CAB. Then if θ is acute, we definesin θ = BCABand cos θ =ACABIf θ is obtuse, then let θ ′ = 180 − θ and definesin θ = sin θ ′ and cos θ = − cos θ ′Also, definesin 0 = 0 and cos 0 = 1sin 90 = 1 and cos 90 = 0Theorem 38.6 (Pythagorean Identity)Proof. (Exercise.)sin 2 θ + cos 2 θ = 1Theorem 38.7 (Law of Sines) Let △ABC be any triangle with sidesa, b, c opposite vertices A, B, C. ThenProof. (Exercise.)asin ∠A =bsin ∠B =csin ∠CTheorem 38.8 (Law of Cosines) Let △ABC be any triangle with sidesa, b, c opposite vertices A, B, C. ThenProof. (Exercise.)c 2 = a 2 + b 2 − 2ab cos ∠CRevised: 18 Nov 2012 « CC BY-NC-ND 3.0.