6.2 The Law of Cosines

2

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

Let's consider types of triangles with the three pieces of information shown below.

SAS

You may have a side, an angle, and then another side

AAA

You may have all three angles.

SSS

You may have all three sides

This case doesn't determine a triangle because similar triangles have the same angles and shape but "blown up" or "shrunk down"

We can't use the Law of Sines on these because we don't have an angle and a side opposite it. We need another method for SAS and SSS triangles.

AAA

LAW OF COSINES

Cabbac cos2222

Baccab cos2222

Abccba cos2222 LAW OF COSINES

ab

cbaC

2cos

222

ac

bcaB

2cos

222

bc

acbA

2cos

222

Use these to findmissing sides

Use these to find missing angles

Do you see a pattern?

5

Deriving the Law of Cosines

• Write an equationusing Pythagorean theorem for shaded triangle.

b h a

k c - kA B

C

c

sin

cos

h b A

k b A

2 22

2 2 2 2 2 2

2 2 2 2 2

2 2 2

sin cos

sin 2 cos cos

sin cos 2 cos

2 cos

a b A c b A

a b A c c b A b A

a b A A c c b A

a b c c b A

Since the Law of Cosines is more involved than the Law of Sines, when you see a triangle to solve you first look to see if you have an angle (or can find one) and a side opposite it. You can do this for ASA, AAS and SSA. In these cases you'd solve using the Law of Sines. However, if the 3 pieces of info you know don't include an angle and side opposite it, you must use the Law of Cosines. These would be for SAS and SSS (remember you can't solve for AAA).

Since the Law of Cosines is more involved than the Law of Sines, when you see a triangle to solve you first look to see if you have an angle (or can find one) and a side opposite it. You can do this for ASA, AAS and SSA. In these cases you'd solve using the Law of Sines. However, if the 3 pieces of info you know don't include an angle and side opposite it, you must use the Law of Cosines. These would be for SAS and SSS (remember you can't solve for AAA).

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

Draw a picture.

80

B

C

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

Abccba cos2222 times the cosine of the angle between

those sides

One side squared

2a

sum of each of the other sides

squared

minus 2 times the productof those

other sides

312 80cos22 31

Now punch buttons on your calculator to find a. It will be square root of right hand side.

a = 2.99

CAUTION: Don't forget order of operations: powers then multiplication BEFORE addition and subtraction

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

B

C

2.99

1

3

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

80.77

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

19.23

1

sin

99.2

80sin B

77.8099.2

80sin3B

23.197.8080180

85.1815.81 andor If you found C first

Cabbac cos2222

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

Draw a picture. B

C

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

times the cosine of the angle between

those sides

One side squared

29

sum of each of the other sides

squared

minus 2 times the productof those

other sides

852 Ccos22 85 CAUTION: Don't forget order of operations: powers then multiplication BEFORE addition and subtraction

A

Ccos808981

80

8cos

C26.84

10

1cos 1

C

84.26

How can we find one of the remaining angles?

B

5

8

9Do we know an angle and side opposite it?

A 84.26

62.18

33.56

Yes, so use Law of Sines.

56.3318.6226.84180 A

8

sin

9

26.84sin B

Bsin9

26.84sin8 18.62

9

26.84sin8sin 1

11

Try it on your own! #1

• Find the three angles of the triangle ABC if

86

A B

C

28.117,34.36,38.26 CBA

12,8,6 cba

12

12

Try it on your own! #2

• Find the remaining angles and side of the triangle ABC if

16

80A B

C

33.40,67.59,26.18 CBa

80,12,16 Amcb

12

13

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)?

• Note these are the actual dimensions!

A

C

14

Wing Span

A

C

Baccab cos2222

05.109cos)6.105)(6.105(26.1056.105 222 b

46.727972.223022 b

.172 ftb

H Dub

• 6-2 Pg. 443 #2-16even, 17-22all, and 29

More Practice #1

More Practice #2