Law of Sines and Cosines (I)

I. Law of Sines and Cosines (Ch.6) I. LAW OF SINES: definition [a/sin A = b/sin B = c/sin C] C a A C b the law of sines is used to solve an oblique triangle, or a triangle that is not right Furthermore, the law of sines can only be used in AAS, ASA, and SSA (ambiguous) cases. ambiguous SSA cases no solutions occurs when sin x > 1 two solutions occurs when angle given is smaller than the angle found subtract angle found from 180° one solution → otherwise II. EXAMPLES Given ΔABC, with m∠A=70°, m∠B=80°, and side a=12m, find m∠C and the side lengths of b and c. C \[sin A / 5 = sin 78 / 3\] \[b = 3\] C \[70° + 80° + c = 180°\] \[a = 5\] 30 \[sin A = 1.6\] \[C = 30°\] b \[a = 12m\] \[\underline{78°}\] B A \[A ∉ R\] \[\frac{b}{sin80} = \frac{12m}{sin70}\] 70 80 A B \[\frac{c}{sin 30} = \frac{12m}{sin 70}\] \[b = 12.6 m\] \[c = 6.4m\] Δ ABC: a² = b² + c² - 2bc cos A b² = a² + c² - 2ac cos B c² = a² + b² - 2ab cos C the law of cosines is used to solve an oblique Furthermore, the law of cosines is only applicable in SSS and SAS cases m∠B = 98°, c = 20 m, a = 15 m, find b. b² = 169 + 400 - 2(13)(20)cos98 b² = 641.4 b ≈ 25.3 Find all angles in Δ ABC: a = 14 cm, b = 9 cm, c = 6 cm. 196 = 81 + 36 - 2(54)cos A 79 = -108 cos A m∠A ≈ 137° sin B / 9 cm = sin 137° / 14 cm m∠B ≈ 26° 137° + 26° + m∠C = 180° m∠C ≈ 17°