The Law of Sines

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The Law of Sines
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Law of Sines

• If A, B, and C are the measures of the angles of
  a triangle, and a, b, and c are the lengths of
  the sides opposite these angles, then
• The ratio of the length of the side of any
  triangle to the sine of the angle opposite that
  side is the same for all three sides of the
  triangle.

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Text Example

• Solve triangle ABC if A 50º, C 33.5º, and b
  76.

Solution We begin by drawing a picture of
triangle ABC and labeling it with the given
information. The figure shows the triangle that
we must solve. We begin by finding B.
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Solve triangle ABC if A 50º, C 33.5º, and b
76.
Text Example cont.
Solution Keep in mind that we must be given one
of the three ratios to apply the Law of Sines. In
this example, we are given that b 76 and we
found that B 96.5º. Thus, we use the ratio
b/sin B, or 76/sin96.5º, to find the other two
sides. Use the Law of Sines to find a and c.
Find a Find c
The solution is B 96.5º, a ? 59, and c ? 42.
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Example

• Solve the triangle shown with A36º, B88º and
  c29 feet.

Solution
A36º, B88º so 180-88-3656º C56º
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The Ambiguous Case (SSA)
Consider a triangle in which a, b, and A are
given. This information may result in
No Triangle One Right Triangle
Two Triangles One
Triangle
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One Solution

• Solve the triangle shown with A43º, a81 and
  b62.

149 43 gt 180. Thus there is only 1 solution.
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Two Solutions

• Solve the triangle shown with X40º, x54 and
  z62.

132 40 lt 180, so we have 2 solutions!
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No Solution

• Solve the triangle shown with A75º, a51 and
  b71.

No Solution!
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Area of An Oblique Triangle

• The area of a triangle equals one-half the
  product of the lengths of two sides times the
  sine of their included angle. In the following
  figure, this wording can be expressed by the
  formulas

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Text Example

• Find the area of a triangle having two sides of
  lengths 24 meters and 10 meters and an included
  angle of 62º.

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Example

• Find the area of a triangle having two sides of
  lengths 12 ft. and 20 ft. and an included angle
  of 57º.

Solution