Law of cosines
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Law of Cosines. HOMEWORK: Lesson 12.4/1-14. 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. Solving an SAS Triangle. The Law of Sines was good for

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Law of Cosines

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Law of cosines

Law of Cosines

HOMEWORK: Lesson 12.4/1-14


Who s law is it anyway

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


Solving an sas triangle

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?

15

26°

No side opposite from any angle to get the ratio

12.5


Law of cosines1

Law of Cosines

  • Note the pattern

C

b

a

A B

c


Law of cosines

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


Applying the cosine law

Applying the Cosine Law

  • Now use it to solve the triangle we started with

  • Label sidesand angles

    • Side c first

C

15

26°

12.5

A B

c


Applying the cosine law1

Applying the Cosine Law

  • Now calculate the angles

    • useand solve for B

C

15

26°

12.5

A B

c = 6.65


Applying the cosine law2

Applying the Cosine Law

  • The remaining angledetermined by subtraction

    • 180 – 93.75 – 26 = 60.25

C

15

26°

12.5

A B

c = 6.65


Law of cosines

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


Law of cosines

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.


Law of cosines

9

5

84.3

8


Wing span

Wing Span

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


Law of cosines

C

105.6 ft

x

109.05°

B

105.6 ft

A


Using the cosine law to find area

Using the Cosine Law to Find Area

  • Recall that

  • We can use the value for hto determinethe area

C

b h a

A B

c


Using the cosine law to find area1

Using the Cosine Law to Find Area

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

  • Note the pattern

C

b a

A B

c


Try it out

Try It Out

Determine the area

127°

24m

12m


Law of cosines

Determine the area

Missing angle – 180-42.8-76.3 = 60.9°

Missing side

60.9°

17.9

42.8°

76.3°


Cost of a lot

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.

324

412

516


Law of cosines

324

412

516


Law of cosines

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°


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