**The Laws of Sine and Cosine**

Lauren Curlee and Matilda Chidlaw

The Law of Sine

When to use:

Use any 2 fractions

Must have side and angle opposite

Use with NON-right triangles

a

A

b

B

c

C

Ambiguous Case: Law of Sines

Given: Angle, Side, Side

ASS

Example

Example of How to use Law of Sine:

Given:

m < C = 30

ac = 7

cb = 6

**The Law of Cosine**

When to Use:

**Always need 2 sides and the angle they form!**

Example of Law of Cosine

c

a

b

B

A

C

30

o

c

A

B

7

6

angle

side

side

7

_____

sin(B)

=

6

____

sin(30)

sin(B) = 7(sin(30))

____

6

That comes out to: m<B = 36

o

But because you were given angle, side, side you need to account for the 2 possible triangles since angle B could change.

30

7

6

* still same triangle according to given information, but angle B has changed.

**How to account for 2**

**nd**

**triangle**

**You need to make angle B obtuse, which means you need to take the angle you found before (36) and subtract it from 180.**

Now the angles you have are <A = 30 and the new <B, or B' = 144. These angles added together equal 174, which is less than 180.

This means that it is an ambiguous case and the value for C' = 6.

Now the angles you have are <A = 30 and the new <B, or B' = 144. These angles added together equal 174, which is less than 180.

This means that it is an ambiguous case and the value for C' = 6.

If you were a given a triangle like this and asked to solve you would need to account for both triangles.