Chapter 10: Applications of Trigonometry Vectors

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Chapter 10 Applications of Trigonometry
Vectors

• 10.1 The Law of Sines

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10.1 The Law of Sines

Congruence Axioms Side-Angle-Side (SAS) If two
sides and the included angle of one triangle
are equal, respectively, to two sides and the
included angle of a second triangle, then the
triangles are congruent. Angle-Side-Angle
(ASA) If two angles and the included side of one
triangle are equal, respectively, to two
angles and the included side of a second
triangle, then the triangles are
congruent. Side-Side-Side (SSS) If three sides of
one triangle are equal to three sides of a
second triangle, the triangles are congruent.
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10.1 Data Required for Solving Oblique Triangles

• Case 1 One side and two angles known
• SAA or ASA
• Case 2 Two sides and one angle not included
• between the sides known
• SSA
• This case may lead to more than one solution.
• Case 3 Two sides and one angle included between
• the sides known
• SAS
• Case 4 Three sides are known
• SSS

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10.1 Derivation of the Law of Sines

• Start with an acute or obtuse triangle
• and construct the perpendicular from
• B to side AC. Let h be the height of this
• perpendicular. Then c and a are the
• hypotenuses of right triangle ADB
• and BDC, respectively.

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10.1 The Law of Sines

• In a similar way, by constructing perpendiculars
  from
• other vertices, the following theorem can be
  proven.
• Alternative forms are sometimes convenient to use

Law of Sines In any triangle ABC, with sides a,
b, and c,
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10.1 Using the Law of Sines to Solve a Triangle

• Example Solve triangle ABC if A 32.0, B
  81.8,
• and a 42.9 centimeters.
• Solution Draw the triangle
• and label the known values.
• Because A, B, and a are known,
• we can apply the law of sines involving these
  variables.

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10.1 Using the Law of Sines to Solve a Triangle

• To find C, use the fact that there are 180 in a
  triangle.
• Now we can find c.

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10.1 Using the Law of Sines in an Application
(ASA)

• Example Two stations are on an east-west
• line 110 miles apart. A forest fire is located
• on a bearing of N 42 E from the western
• station at A and a bearing of N 15 E from
• the eastern station at B. How far is the fire
• from the western station?
• Solution Angle BAC 90 42 48
• Angle B 90 15 105
• Angle C 180 105 48 27
• Using the law of sines to find b gives

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10.1 Ambiguous Case

• If given the lengths of two sides and the angle
  opposite one of them, it is possible that 0, 1,
  or 2 such triangles exist.
• Some basic facts that should be kept in mind
• For any angle ?, 1 ? sin ? ? 1, if sin ? 1,
  then
• ? 90 and the triangle is a right triangle.
• sin ? sin(180 ? ).
• The smallest angle is opposite the shortest side,
  the largest angle is opposite the longest side,
  and the middle-value angle is opposite the
  intermediate side (assuming unequal sides).

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10.1 Ambiguous Case
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10.1 Ambiguous Case for Obtuse Angle A

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10.1 Solving the Ambiguous Case No Such Triangle

• Example Solve the triangle ABC if B 5540,
• b 8.94 meters, and a 25.1 meters.
• Solution Use the law of sines to find A.
• Since sin A cannot be greater than 1, the
  triangle does
• not exist.

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10.1 Solving the Ambiguous Case Two Triangles

• Example
• Solve the triangle ABC if A 55.3,
• a 22.8 feet, and b 24.9 feet.
• Solution

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10.1 Solving the Ambiguous Case Two Triangles

• To see if B2 116.1 is a valid possibility, add
  116.1
• to the measure of A 116.1 55.3 171.4.
  Since
• this sum is less than 180, it is a valid
  triangle.
• Now separate the triangles into two AB1C1 and
  AB2C2.

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10.1 Solving the Ambiguous Case Two Triangles

• Now solve for triangle AB2C2.

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10.1 Number of Triangles Satisfying the
Ambiguous Case

• Let sides a and b and angle A be given in
  triangle ABC. (The
• law of sines can be used to calculate sin B.)
• If sin B gt 1, then no triangle satisfies the
  given conditions.
• If sin B 1, then one triangle satisfies the
  given conditions and B 90.
• If 0 lt sin B lt 1, then either one or two
  triangles satisfy the given conditions
• If sin B k, then let B1 sin-1 k and use B1
  for B in the first triangle.
• Let B2 180 B1. If A B2 lt 180, then a
  second triangle exists. In this case, use B2 for
  B in the second triangle.

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10.1 Solving the Ambiguous Case One Triangle

• Example Solve the triangle ABC, given A 43.5,
  
• a 10.7 inches, and c 7.2 inches.
• Solution
• The other possible value for C
• C 180 27.6 152.4.
• Add this to A 152.4 43.5 195.9 gt 180
• Therefore, there can be only one triangle.

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10.1 Solving the Ambiguous Case One Triangle