Chapter 6.1 Law of Sines

Oblique triangle: a triangle with no right angle

To solve an oblique triangle with law of sines one must know:

1) Two angles and any side (AAS or SAS)

or

2) Two sides and an angle opposite one of them (SSA)

The Law of Sines

If ABC is a triangle with sides a, b, and c, then:

(a/sinA)=(b/sinB)=(c/sinC)

The law of sines can also be written in reciprocal form.

Derivation of the law of sines

sinB = h / c

c(sinB) = h

sinC = h / b

sinB = c(sinB)/a

(sinB) / (bsinA) = 1 / a

sinB / b = sinA / a

The Ambiguous Case (SSA)

When one is give a triangle in which a, b, and A(h=bsinA) are given

The ambiguous case can provide 0,1, or 2 solutions using the law of sines because it can create more than 1 triangles.

For example 

OR 

These triangles both have the same 2 side lengths and angle measure, yet they are not congruent triangles.

The area of an oblique triangle can also be found with the law of sines

A=.5bcsinA=.5absinC=.5acsinB