• Work out problems on board

• Add visuals (ranges of arccos and arcsin) to show why you use LOC for big angles and LOS for small angles

6.2 The Law of Cosines

Which proved triangles congruent in Geometry?

• SSS

• ASA

• AAS

• SAS

• AAA

• ASS

The same ones that define a specific triangle!

• SSS - congruent

• ASA - congruent

• AAS – congruent

• SAS – congruent

• AAA – not congruent

• ASS – not congruent

Which proved triangles congruent in Geometry?

• SSS - congruent

• ASA – congruent – Solve w/ Law of Sines

• AAS – congruent – Solve w/ Law of Sines

• SAS – congruent

• AAA – not congruent

• ASS – not congruent

6

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

Do you see a pattern?

9

Deriving the Law of Cosines• Write an equation using Pythagorean

theorem for shaded triangle that

only includes sides and angles of the

oblique 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

222 )( kcha

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).

Ex. 1: 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 = 2.9930

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

2.993

1

3

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

80.79

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

19.21

1

sin

993.2

80sin B

21.19993.2

80sinsin

B

B

79.8021.1980180

When taking arcsin, use 2nd answer on your calculator for accuracy!

Cabbac cos2222

Ex. 2: 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 choose to find angle C 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

C2608.84

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.

5573.331819.622608.84180 A

8

sin

9

26.84sin B

Bsin9

2608.84sin8 1819.62

9

2608.84sin8sin 1

Too easy, what’s the catch?• After we use L.O.C. we need to use law of sines to find

the remaining sides and angles. • The range of arcsin is -90 deg to 90 deg, but what if

the angle is obtuse? Then taking the arcsin won’t get us the correct angle!

• To avoid this problem – When using L.O.S. after L.O.C. always find the smallest angle FIRST The smallest angle has to be acute since there can’t be more than one obtuse angle in a triangle.

• Then use the triangle sum thm to find the 3rd angle.

16

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

17

Try it on your own! #2

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

16

80A B

C

33.40,67.59,257.18 CBa

80,12,16 Amcb

12

18

Summary – What could we use to solve the following triangles?

80

30

70

Uh, nothing. It’s AAA

19

20

80

16

ASA – although we don’t know an angle and side opposite each other we can find the 3rd angle

then do law of sines

Summary – What could we use to solve the following triangles?

Do we know an angle and side

opposite it? Could we find it?

20

80

2016

AAS – law of sines

Summary – What could we use to solve the following triangles?

Do we know an angle and side

opposite it?

21

16

80

20

ASS, we can use law of sines but need to check for 1, 2 or no

triangles.

Summary – What could we use to solve the following triangles?

Do we know an angle and side

opposite it?

22

16

8012

SAS – don’t know (and can’t find) angle and side opposite

Law of Cosines

Summary – What could we use to solve the following triangles?

Do we know an angle and side

opposite it?

23

16

12

20

SSS – don’t know (and can’t find) angle and side opposite

Law of Cosines

Summary – What could we use to solve the following triangles?

Do we know an angle and side

opposite it?

24

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

25

Wing Span

A

C

Baccab cos2222

05.109cos)6.105)(6.105(26.1056.105 222 b

46.727972.223022 b

.172 ftb

Navigational Bearings

• The direction to a point is stated as the number of degrees east or west of north or south. For example, the direction of

• A from O is N30ºE.B is N60ºW from O.C is S70ºE from O.D is S80ºW from O

H Dub

• 6-2 Pg. 443 #2-16even, 17-22all, 29, 34, 35

Practice #1

Practice #2

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?

Practice #1

Practice #2