Law of Cosines

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Law of Cosines

• Day 67

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Plan for the day

• Review Homework 6.1 page 416 3, 5, 19-24 all,
  25-37 odd
• Law of Cosines
• Finding the area of a triangle Herons Formula
• Homework
• 6.1 page 417 39
• 6.2 page 423 5-11 odd, 23-27 odd,31,33

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What You Should Learn

• Use the Law of Cosines to solve oblique triangles
  (SSS or SAS).
• Use the Law of Cosines to model and solve
  real-life problems.
• Use Herons Area Formula to find the area of a
  triangle.

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Introduction

• Four cases.
• Two angles and any side (AAS or ASA)
• Two sides and an angle opposite one of them (SSA)
• Three sides (SSS)
• Two sides and their included angle (SAS)
• The first two cases can be solved using the Law
  of Sines, whereas the last two cases require the
  Law of Cosines.

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Law of Sines

• For non right triangles
• Law of sines

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Ambiguous Case

1. Is it Law or Sines or Law of Cosines
2. Law of Cosines solve based upon one solution
3. Law of Sines go to 2
4. Law of Sines - Is it the SSA case? (Two sides
   and angle opposite)
5. No not ambiguous, solve based upon one solution
6. Yes go to 3.
7. Is the side opposite the angle the shortest side?
8. No not ambiguous, solve based upon one solution
9. Yes go to 4
10. Is the angle obtuse?
11. No go to 5
12. Yes no solution
13. Calculate the height of the triangleheight the
    side not opposite the angle x the sine of the
    angle
14. If the side opposite the angle is shorter than
    the height no solution
15. If the side opposite the angle is equal to the
    height one solution
16. If the side opposite the angle is longer than the
    height two solutions

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Area of a Triangle - SAS

• SAS you know two sides b, c and the angle
  between A
• Remember area of a triangle is ½ base ? height
• Base b
• Height c ? sin A
• ? Area ½ bc(sinA)

Looking at this from all three sides Area ½
ab(sin C) ½ ac(sin B) ½ bc(sin A)
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Law of Cosines Introduction

• Two cases remain in the list of conditions needed
  to solve an oblique triangle SSS and SAS.
• If you are given three sides (SSS), or two sides
  and their included angle (SAS), none of the
  ratios in the Law of Sines would be complete.
• In such cases, you can use the Law of Cosines.

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Law of Cosines

• Side, Angle, Side

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Try these

• B 20o a 10 c 15
• A 25o b 3 c 6
• C 75o a 5 b 3

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Law of Cosines

• Side, Side, Side

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Law of Cosines

• SSS

• Always solve for the angle across from the
  longest side first!

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Why

• It is wise to find the largest angle when you
  have SSS. Knowing the cosine of an angle, you can
  determine whether the angle is acute or obtuse.
  That is,
• cos ? gt 0 for 0? lt ? lt 90?
• cos ? lt 0 for 90? lt ? lt 180?.
• This avoids the ambiguous case!

Acute
Obtuse
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Try these

1. a 5 b 4 c 6
2. a 20 b 10 c 28
3. a 8 b 5 c 12

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Applications
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An Application of the Law of Cosines

• The pitchers mound on a womens softball field
  is 43 feet from home plate and the distance
  between the bases is 60 feet (The pitchers mound
  is not halfway between home plate and second
  base.) How far is the pitchers mound from first
  base?

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Solution

• In triangle HPF, H 45? (line HP bisects the
  right angle at H), f 43, and p 60.
• Using the Law of Cosines for this SAS case, you
  have
• h2 f 2 p2 2fp cos H
• 432 602 2(43)(60) cos 45?
  ? 1800.3.
• So, the approximate distance from the pitchers
  mound to first base is ?
  42.43 feet.

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Herons Formula
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Herons Area Formula

• The Law of Cosines can be used to establish the
  following formula for the area of a triangle.
  This formula is called Herons Area Formula after
  the Greek mathematician Heron (c. 100 B.C.).

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Area of a Triangle Law of Cosines Case - SSS
SSS Given all three sides
Herons formula
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Try these

• Given the triangle with three sides of 6, 8, 10
  find the area
• Given the triangle with three sides of 12, 15, 21
  find the area

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Homework 36

• 6.1 page 417 39
• 6.2 page 423 5-13 odd, 23-27 odd, 31, 33