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Law of Cosines. Lesson 4.2. 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. Solving an SAS Triangle. The Law of Sines was good for ASA - two angles and the included side

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Law of cosines

Law of Cosines

Lesson 4.2


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


Deriving the law of cosines
Deriving the Law of Cosines

C

  • Write an equationusing Pythagorean theorem for shaded triangle.

b h a

k c - k

A B

c


Law of cosines1
Law of Cosines

  • Similarly

  • Note the pattern


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

Experiment with Cosine Law Spreadsheet


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


Assignment a
Assignment A

  • Lesson 4.2A

  • Page 308

  • Exercises 1 – 27 odd, and 41 - 51 odd


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 of these triangles

42.8°

17.9

127°

24

12

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


Assignment b
Assignment B

  • Lesson 4.2B

  • Page 309

  • Exercises 29 – 39 oddand 53 – 61 odd


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