Law of Cosines

HOMEWORK: Lesson 12.4/1-14

2

Who's Law Is It, Anyway?

Murphy's Law: Anything that can possibly go wrong, will

go wrong (at the worst possible moment). Cole's Law ??

Finely chopped cabbage

3

Solving an SAS Triangle

The Law of Sines was good for ASA - two angles and the included side AAS - two angles and any side SSA - two sides and an opposite angle

(being aware of possible ambiguity)

Why would the Law of Sines not work for an SAS triangle?

1512.5

26°No side opposite from any angle to

get the ratio

No side opposite from any angle to

get the ratio

4

Law of Cosines

Note the pattern

2 2 2

2 2 2

2 2 2

2 cos

2 cos

2 cos

a b c c b A

b a c a c B

c b a a b C

A B

C

a

c

b

We could do the same thing if gamma was obtuse and we could repeat this process for each of the other sides. We end up with the following:

LAW OF COSINES

cos2222 abbac

cos2222 accab

cos2222 bccba LAW OF COSINES

ab

cba

2cos

222

ac

bca

2cos

222

bc

acb

2cos

222

Use these to findmissing sides

Use these to find missing angles

6

Applying the Cosine Law

Now use it to solve the triangle we started with

Label sidesand angles Side c first

1512.5

26°

A B

C

c

2 2 2

2 2

2 cos

15 12.5 2 15 12.5 cos 26

c b a a b C

c

7

Applying the Cosine Law

Now calculate the angles use

and solve for B

1512.5

26°

A B

C

c = 6.65

2 2 2 2 cosb a c a c B

2 2 2 2 2 21cos cos

2 2

b a c b a cB B

a c a c

8

Applying the Cosine Law

The remaining angledetermined by subtraction 180 – 93.75 – 26 = 60.25

1512.5

26°

A B

C

c = 6.65

Solve a triangle where b = 1, c = 3 and = 80°

Draw a picture.

80

a

1

3

Do we know an angle and side opposite it? No so we must use Law of Cosines.

Hint: we will be solving for the side opposite the angle we know.

This is SAS

cos2222 bccba

2a 312 80cos22 31 a = 2.99

cos2222 abbac

Solve a triangle where a = 5, b = 8 and c = 9

Draw a picture.

5

8

9

Do we know an angle and side opposite it? No, so we must use Law of Cosines.

Let's use largest side to find largest angle first.

This is SSS

29 852 cos22 85

cos808981

80

8cos

3.8410

1cos 1

84.3

11

5

8

9

84.3

12

Wing Span

The leading edge ofeach wing of theB-2 Stealth Bombermeasures 105.6 feetin length. The angle between the wing's leading edges is 109.05°. What is the wing span (the distance from A to C)?

Hint … use the law of cosines!

A

C

13

105.6 ft

109.05°

C

A

x

B

2 2 2 2 cosb a c a c B

105.6

ft

14

Using the Cosine Law to Find Area

Recall that

We can use the value for hto determinethe area

b h a

A B

c

sinh b A

1sin

2Area c b A

C

15

Using the Cosine Law to Find Area

We can find the area knowing two sides and the included angle

Note the pattern

1sin

21

sin2

Area a b C

c a B

b a

A Bc

C

Determine the area

16

Try It Out

127°12m 24m

17

76.3

°

42.8°17.9

Determine the area

Missing angle – 180-42.8-76.3 = 60.9°

60.9°

a

9.60sin

9.17

3.76sin

Missing side

18

Cost of a Lot An industrial piece of real estate is

priced at $4.15 per square foot. Find, to the nearest $1000, the cost of a triangular lot measuring 324 feet by 516 feet by 412 feet.

516

412

324

19

516

412

324

We'll label side a with the value we found.

We now have all of the sides but how can we find an angle?

80

2.99

1

3

Hint: We have an angle and a side opposite it.

sin80 sin

2.99 3

3sin80

2.99 80.8

80.8

is easy to find since the sum of the angles is a triangle is 180°

180 80 80.8 19.2

19.2