When can we use the Sine Law?

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When can we use the Sine Law? B A a C c b SINE LAW : 𝑎 𝑠𝑖𝑛𝐴 = 𝑏 𝑠𝑖𝑛𝐵 = 𝑐 𝑠𝑖𝑛𝐶 So if we knew Angles A and B and side c our calculations would be: Case 1: Two Angles & a Side ( AAS) solve for the 3rd unknown angle using the fact that the 3 angles add to 180o Use 2 ratios of the Sine Law to solve for one of the two unknown sides Use 2 ratios of the Sine Law to solve for the last unknown side 𝐶= 180 𝑜 −𝐴−𝐵 𝑎 𝑠𝑖𝑛𝐴 = 𝑐 𝑠𝑖𝑛𝐶 ⇒𝑎= 𝑐(𝑠𝑖𝑛𝐴) 𝑠𝑖𝑛𝐶 𝑏 𝑠𝑖𝑛𝐵 = 𝑐 𝑠𝑖𝑛𝐶 ⇒𝑏= 𝑐(𝑠𝑖𝑛𝐵) 𝑠𝑖𝑛𝐶 Washington/Evans Basic Technical Mathematics 11e -- Copyright © 2018 Pearson Inc.

When can we use the Sine Law? B b c SINE LAW : 𝑎 𝑠𝑖𝑛𝐴 = 𝑏 𝑠𝑖𝑛𝐵 = 𝑐 𝑠𝑖𝑛𝐶 Case 2: Two Sides & Angle Opposite a Side ( SSAopp) solve for the unknown angle opposite the known side solve for the last unknown angle using the fact that the 3 angles add to 180o Use 2 ratios of the Sine Law to solve for the last unknown sides So if we knew Angle A and sides a and b our calculations would be: 𝑎 𝑠𝑖𝑛𝐴 = 𝑏 𝑠𝑖𝑛𝐵 ⇒𝑠𝑖𝑛𝐵= 𝑏(𝑠𝑖𝑛𝐴) 𝑎 𝐶= 180 𝑜 −𝐴−𝐵 𝑎 𝑠𝑖𝑛𝐴 = 𝑐 𝑠𝑖𝑛𝐶 ⇒𝑐= 𝑎(𝑠𝑖𝑛𝐶) 𝑠𝑖𝑛𝐴 Washington/Evans Basic Technical Mathematics 11e -- Copyright © 2018 Pearson Inc.

Case 2 – possible solutions SINE LAW : 𝑎 𝑠𝑖𝑛𝐴 = 𝑏 𝑠𝑖𝑛𝐵 = 𝑐 𝑠𝑖𝑛𝐶 Case 2: Two Sides & Angle Opposite a Side ( SSAopp) we could get the following solutions: No solution if 𝑎<𝑏(𝑠𝑖𝑛𝐴) a b A A a b A right triangle solution if 𝑎=𝑏 𝑠𝑖𝑛𝐴 Called Ambiguous Case a b A B C a b A C’ B’ a b A Two solutions if 𝑏(𝑠𝑖𝑛𝐴)<𝑎<𝑏 One solution if 𝑎>𝑏(𝑠𝑖𝑛𝐴) Washington/Evans Basic Technical Mathematics 11e -- Copyright © 2018 Pearson Inc.