There is no doubt that the 3 PTRs are extremely useful when solving problems modeled on a right triangle.

Unfortunately, the world does not consist only of right triangles…

As a matter of fact, right triangles end up being more of a rarity than commonplace.

There are many situations where angles other than 90O are present.

Does that mean when we come across a situation that can only be modeled with a non-right triangle that we abandon our pursuit?….

No Way!!!!There exists two Laws of

Trigonometry that allow one to solve problems that involve non-right Triangles:

RememberA capital letter represents an

angle in a triangle, and a small letter represents a side of a

triangle

A

a

A BO

C

b

c

a

If b,c and O are all known, then O is called a “Contained Angle”

(the blue line also forms a “c”, [kind of] which is how I remember to use the “c”osine law in this case..)

A BO

C

b

c

a

The Cosine Law can be used to find the length of the opposite side to O

In this case, the length of side a

In General:

a2 = b2 + c2 – 2bcCosOo

A BO

C

b

c

a

For Example: Find a

A50o

C

8m

10m

a

B

a2 = 82 + 102 – 2(8)(10)Cos50o

A50o

C

8m

10m

a

a2 = 61.15m a = 7.8 m

B

a2 = b2 + c2 – 2(b)(c)CosAo

You should be able to load this into your calculator directly from left to right…if not, see me

The Sine Law

If the triangle being solved does not consists of a right triangle (3PTRs) or a contained angle (Cosine Law), then another tool must be used.

If a corresponding angle and side are known, they form an

“opposing pair”

A

C

Bc

b a

O1 O2

The Sine Law can be used to determine an unknown side or angle given an “opposing pair”

A

C

Bc

b a

O1 O2

The Sine Law

b SinA = SinB

a c = SinC

A

C

Bc

ba

Find the length of a

a

A

C

c

2473o

57o

N

We can not use the Cosine Law because there is not a contained angle…

We must therefore look for an opposite pair. Hmmm…..

A-HA!!!(it’s all good)

Find the length of a

a

A

C

c

24

57o

N

aSin73o

= 24Sin57o

a = 27.4

73o

Again, this can be put directly into your calculator. See me for help.

Pg 2901a,c,d,e4a,c,e5a,c68,10,12,14Pg 2951(11 unco, stop

here)

The Ambiguous Case

Find A

11

9

48oA

11

SinA =9

Sin48o

A = 65.3o

Does that make sense?

No Way!!!

Side 9 can also be drawn as:

119

48o

A

Could A be 65o in this

case?

This type of discrepancy is called the “Ambiguous Case”

Be sure to check the diagram to see which answer fits:

O, or 180o - O