Law Of Sines

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Law Of Sines

• Section 7.1
• 9/17/08

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7.1 The Law of Sines

• OBJECTIVES
• Use the law of sines to solve triangles.
• Find the area of any triangle given the lengths
  of two sides and the measure of the included
  angle.

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Solving Oblique Triangles

• To solve a triangle means to find the lengths of
  all its sides and the measures of all its angles.
• Any triangle that is not a right triangle is
  called oblique.
• Any triangle, right or oblique, can be solved if
  at least one side and any other two measures are
  known.

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5 Possible Triangles
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The Law of Sines

• In any triangle ABC,
• Thus in any triangle, the sides are proportional
  to the sines of the opposite angles.

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Solving a Triangle (AAS)

• In triangle ABC, A 57?, B 43?, and b 11.2.
  Solve the triangle.

• C (180 ? (57 43))
• C 180 ? 100 80
• Find a and c by using the law of sines

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Solving a Triangle (SSA)

• If the given angle is acute, then there may be
• a) no solution,
• b) one solution, or
• c) two solutions
• If the given angle is obtuse, then there may be
• a) no solution, or
• b) one solution

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Example

• In triangle ABC, b 8.6, c 6.2, and C 35?.
  Solve the triangle.
• Solution

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Example continued

• There are two solutions

or
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The Area of a Triangle

• The area K of any ?ABC is one half the product of
  the lengths of two sides and the sine of the
  included angle

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Example

• Find the area of the triangle, ABC with A 72?,
  b 16, and c 10.
• Solution