Chapter 10: Applications of Trigonometry Vectors

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Chapter 10 Applications of Trigonometry
Vectors

• 10.2 The Law of Cosines and Area Formulas

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10.2 The Law of Cosines and Area Formulas

• SAS or SSS forms a unique triangle
• Triangle Side Restriction
• In any triangle, the sum of the lengths of any
  two sides must be greater than the length of the
  remaining side.

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10.2 Derivation of the Law of Cosines

• Let ABC be any oblique triangle drawn with its
  vertices
• labeled as in the figure below.
• The coordinates of point A become (c cos B, c sin
  B).

Figure 10 pg 10-28
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10.2 Derivation of the Law of Cosines

• Point C has coordinates (a, 0) and AC has length
  b.
• This result is one form of the law of cosines.
  Placing A
• or C at the origin would have given the same
  result, but
• with the variables rearranged.

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10.2 The Law of Cosines
The Law of Cosines In any triangle ABC, with
sides a, b, and c,
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10.2 Using the Law of Cosines to Solve a
Triangle (SAS)

• Example Solve triangle ABC if
• A 42.3, b 12.9 meters, and
• c 15.4 meters.
• Solution Start by finding a using the law of
  cosines.

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10.2 Using the Law of Cosines to Solve a
Triangle (SAS)

• B must be the smaller of the two remaining angles
  
• since it is opposite the shorter of the two sides
  b and c.
• Therefore, it cannot be obtuse.

Caution If we had chosen to find C rather than
B, we would not have known whether C equals 81.7
or its supplement, 98.3.
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10.2 Using the Law of Cosines to Solve a
Triangle (SSS)

• Example Solve triangle ABC if a 9.47 feet,
• b 15.9 feet, and c 21.1 feet.
• Solution We solve for C, the largest angle,
  first. If
• cos C lt 0, then C will be obtuse.

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10.2 Using the Law of Cosines to Solve a
Triangle (SSS)

• Verify with either the law of sines or the law of
  cosines
• that B ? 45.1. Then,

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10.2 Summary of Cases with Suggested Procedures

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10.2 Summary of Cases with Suggested Procedures

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10.2 Area Formulas

• The law of cosines can be used to derive a
  formula for the area of a triangle given the
  lengths of three sides known as Herons Formula.

Herons Formula If a triangle has sides of
lengths a, b, and c, and if the semiperimeter
is Then the area of the triangle is
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10.2 Using Herons Formula to Find an Area

• Example The distance as the crow flies from Los
  Angeles
• to New York is 2451 miles, from New York to
  Montreal is 331
• miles, and from Montreal to Los Angeles is 2427
  miles. What is
• the area of the triangular region having these
  three cities as
• vertices? (Ignore the curvature of the earth.)
• Solution

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10.2 Area of a Triangle Given SAS

• The area of any triangle is given by A ½bh,
  where b is its base and h is its height.

Area of a Triangle In any triangle ABC, the area
A is given by any of the following
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10.2 Finding the Area of a Triangle (SAS)

• Example Find the area of triangle ABC in the
  figure.
• Solution We are given B 55, a 34 feet, and
• c 42 feet.