The Law of Sines

2.3 I can solve triangles using the Law of Sines

If none of the angles of a triangle is a right angle, the triangle is called oblique. All angles are acute Two acute angles, one obtuse angle

To solve an oblique triangle means to find the lengths of its sides and the measurements of its angles.

FOUR CASES CASE 1: One side and two angles are known (SAA or ASA). CASE 2: Two sides and the angle opposite one of them are known (SSA). Ambiguous Case CASE 3: Two sides and the included angle are known (SAS). CASE 4: Three sides are known (SSS).

A S A ASA CASE 1: ASA or SAA Use Law of Sines S A A SAA

S A S CASE 2: SSA - Ambiguous Case Use Law of Sines

S A S CASE 3: SAS Use Law of Cosine

S S S CASE 4: SSS Use Law of Cosines

Theorem Law of Sines

Case 1

Case 1

The Ambiguous Case: Case 2: SSA The known information may result in One triangle Two triangles No triangles

Case 2 Not possible, so there is only one triangle!

Case 2 Two triangles!!

Triangle 1:

Triangle 2:

No triangle with the given measurements! Case 2 No triangle with the given measurements!

The Ambiguous Case: Case 2: SSA The known information may result in One triangle Two triangles No triangles The key to determining the possible triangles, if any, lies primarily with the height, h and the fact h = b sin α a b h α

No Triangle If a < h = b sin α, then side a is not sufficiently long to form a triangle. a < h = b sin α b a h = b sinα α

One Right Triangle If a = h = b sin α, then side a is just long enough to form a triangle. a = h = b sin α b a h = b sinα α

Two Triangles If a < b and h = b sin α < a, then two distinct triangles can be formed a < b and h = b sin α < a a b a h = b sinα α

One Triangle If a ≥ b, then only one triangle can be formed. a ≥ b Fortunately we do not have to rely on the illustration to draw a correct conclusion. The Law of Sines will help us. a b h = b sinα α