Using the Law of Sines

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Using the Law of Sines

• What you will learn To use the law of sines to
  solve triangles
• Why its important You can use the law of sines
  to find missing measures of triangles involved in
  surveying, aviation, and fire fighting

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Law of Sines

• Let ABC be any triangle with a, b, and c
  representing the measures of sides opposite
  angles with measures A, B, and C, respectively.
  Then, sin A sin B sin C.
• a b c

C
b
a
A
B
c
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Proof of Law of Sines

• ABC is a triangle with an altitude from c
  that intersects AB at D. Let h represent the
  measure of CD. Since ACD and BCD are
  right triangles, we can find sin A and sin B.

C
b
a
h
A
D
B
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Solve the triangle

• Finding the measures of all the angles and any
  sides of a triangle is called solving the
  triangle. The Law of Sines can be used to solve a
  triangle in the following cases.
• You know the measures of two angles and any side
  of a triangle.
• You know the measures of two sides and an angle
  opposite one of these sides of the triangle.

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Example 1

• Solve XYZ if m X33, m Z47, and z 14.
  since the measures of two angles and a side are
  known, this is and example of Case 1 Consider the
  four parts angle X, angle Z, x, and z

X
Z
Y
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Example 2

• Anna Garcia is a surveyor who has the job of
  determining the distance across the Rio Grande
  Gorge in northern New Mexico. Standing at one
  side of the ridge, she measures the angle formed
  by the edge of the ridge and the line of sight to
  a tree on the other side of the ridge. She then
  walks along the ridge 315 feet north and measures
  the angle formed by the edge of the ridge and the
  new line of sight to the same tree. If the first
  angle 80 and the second angles is 85, find the
  distance across the gorge.

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Assignment Page 42911 31 odd, 33 40 all