1. 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.

2. 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.

3. 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.