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Law of Cosines

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Law of Cosines

HOMEWORK: Lesson 12.4/1-14

- Murphy's Law:
- Anything that can possibly go wrong, will go wrong (at the worst possible moment).

- Cole's Law ??
- Finely chopped cabbage

- 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?

15

26°

No side opposite from any angle to get the ratio

12.5

- Note the pattern

C

b

a

A B

c

LAW OF COSINES

LAW OF COSINES

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:

Use these to findmissing sides

Use these to find missing angles

- Now use it to solve the triangle we started with
- Label sidesand angles
- Side c first

C

15

26°

12.5

A B

c

- Now calculate the angles
- useand solve for B

C

15

26°

12.5

A B

c = 6.65

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

C

15

26°

12.5

A B

c = 6.65

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

Draw a picture.

This is SAS

3

a

80

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

1

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

a = 2.99

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

Draw a picture.

This is SSS

9

5

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

84.3

8

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

9

5

84.3

8

C

- 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

105.6 ft

x

109.05°

B

105.6 ft

A

- Recall that
- We can use the value for hto determinethe area

C

b h a

A B

c

- We can find the area knowing two sides and the included angle
- Note the pattern

C

b a

A B

c

Determine the area

127°

24m

12m

Determine the area

Missing angle – 180-42.8-76.3 = 60.9°

Missing side

60.9°

17.9

42.8°

76.3°

- 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.

324

412

516

324

412

516

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

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

3

19.2

2.99

80.8

80

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

1

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