General CommentsWe learned to solve right

triangles in section 2. You will be learning to solve oblique triangles (non-right triangles).

Please note that angles are Capital letters and the side opposite is the same letter in lower case.

A

C

B

ab

AB

C

ab

c

c

What we already know The interior angles total 180.

We can’t use the Pythagorean Theorem. Why not?

Area = ½ bh

AB

C

ab

c

Playing with a the triangle

AB

C

ab

c

Let’s drop an altitude

and call it h.

h

If we think of h as

being opposite to

both A and B, then

sin sinh h

A and Bb a

Let’s solve both for h.

sin sinh b A and h a B

This means

sin sin and dividing by .

sinA sin

a

b A a B ab

B

b

AB

C

ab

c

If I were to drop an altitude to

side a, I could come up with

sin sinB C

b c

Putting it all together gives us

the Law of Sines.

sin sin sinA B C

a b c

You can also use it

upside-down. sin sin sin

a b c

A B C

What good is it?

The Law of Sines can be used to solve the following types of

oblique triangles

•Triangles with 2 known angles and 1 side (AAS or ASA)

•Triangles with 2 known sides and 1 angle opposite one of the

sides (SSA)

With these types of triangles, you will almost always have

enough information/data to fill out one of the fractions.

sin sin sinA B C

a b c

General Process1. Except for the ASA triangle, you will always have

enough information for 1 full fraction and half of another. Start with that to find a fourth piece of data.

2. Once you know 2 angles, you can subtract from 180 to find the 3rd.

3. To avoid rounding error, use given data instead of computed data whenever possible.

Example 1: AAS

AB

C

ab

c

45 , 50 , 30Let A B a

Once I have 2 angles, I can

find the missing angle by

subtracting from 180.

C=180 – 45 – 50 = 85

45 50

=3085 I’m given both pieces for

sinA/a and part of sinB/b,

so we start there.

sin45 sin50

30 b

Cross multiply

and divide to get

sin 45 30sin50

30sin50

sin 45

b

b

45 50

=3085

AB

C

ab

c

30sin50

sin45b

Using a calculator,

b 32.5

b 32.5

We’ll repeat the process to find

side c.

Remember to avoid rounded

values when computing.

sin sin

sin 45 sin85

30

A C

a c

c

Cross multiply and divide to

get

sin 45 30sin85

30sin85

sin 45

c

c

Using a calculator,

42.3c

c 42.3

We’re done when we

know all 3 sides and all 3

angles.

Example 2: ASA

AB

C

ab

c

Let A = 35, B = 10, and c = 45

35 10

45

Since we can’t start one of the

fractions, we’ll start by finding C.

C = 180 – 35 – 10 = 135

135

Since the angles were exact, this

isn’t a rounded value. We use

sinC/c as our starting fraction.

sin sin sin sinC A C Band

c a c b

sin135 sin35

45 a

Cross multiply and divide

sin135 45sin 35

45sin 35

sin135

a

a

Using your calculator

36.5a

36.5

sin135 sin10

45 b

sin135 45sin10

45sin10

sin135

b

b

11.1b

11.1

Example 3: SSA

AB

C

ab

c

Let A = 40, b = 10, and a = 7

We have enough information

to use sinA/a as our starting

fraction and can go to work on

B

sin sin

sin 40 sin

7 10

A B

a b

B

We cross multiply and

divide to get

10sin40sin

7B

We have to get rid of the sin function to find

the answer for angle B.

To do this, we apply the inverse operation.

1

1

1 10sin 40sin

7s

10sin 40sin

7

in sinB

B

With a calculator

66.7B

40

10 7

66.7

40

10 7

66.7

AB

C

ab

c

We need something to start the fraction

sinC/c. Since it is critical that the

angles total 180, we will find C by

subtracting from 180.

C=180 – 40 – 66.7 = 73.3

73.3

Now we use our starting fraction,

sinA/a with sinC/c.

sin40 sin73.3

7 c

Cross multiply and divide to

finish.

sin 40 7sin 73.3

7sin 73.310.4

sin 40

c

c

10.4

Remember:

The New Way to Find AreaWe all know that A = ½ bh.

And a few slides back we

found this.

sin sinh b A and h a B AB

C

ab

c

It looks like the base is c.

Making a little substitution

gives me

1 1sin sin

2 2Area bc A ac B

If we drop the altitude in a

different direction, we can

add

1sin

2Area ab C

It’s always the 2 sides and the

included angle.

An Area ExampleFind the area of a triangle with side a = 10, side b =

12, and angle C = 40.

1. Choose the appropriate area formula.

In this case we choose Area = ½ ab sinC.

2. Fill in the blanks.

Area = ½ (10)(12)sin40

3. Plug it into your calculator

Area = 38.6 square units