Law of Sines

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Law of Sines

• MATH 109 - Precalculus
• S. Rook

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Overview

• Section 6.1 in the textbook
• Law of Sines

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Law of Sines
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Oblique Triangles

• Oblique Triangle a triangle containing no right
  angles
• All of the triangles we have studied thus far
  have been right triangles
• We can apply SOHCAHTOA only to right triangles
• Naturally most triangles will not be right
  triangles
• Thus we need a method to apply to find side
  lengths and angles of other types of triangles

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Four Cases for Oblique Triangles

• By definition a triangle has three sides and
  three angles for a total of 6 components
• We can find the measure of all sides and all
  angles if we know AT LEAST 3 of these components
• Broken down into four cases
• AAS or ASA
• Measure of two angles and the length of any side
• SSA
• Length of two sides and the measure of the angle
  opposite one of two known sides
• Known as the ambiguous case because none, one, or
  two triangles could be possible

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Four Cases for Oblique Triangles (Continued)

• SSS
• Length of all three sides
• SAS
• Length of two sides and the measure of the angle
  opposite the third (possibly unknown) side
• AAA is NOT a case because there are an infinite
  number of triangles that can be drawn
• Recall that the largest side is opposite the
  largest and angle, but there is no limitation on
  the length of the side!

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Law of Sines

• The first two cases (AAS/ASA and SSA) are covered
  by the Law of Sines
• i.e. the ratio of the measure of any side of a
  triangle to its corresponding angle yields the
  same constant value
• This constant value is different for each
  triangle
• The Law of Sines can be proved by dropping an
  altitude from an oblique triangle and using
  trigonometric functions with the right angle
• See page 489
• ALWAYS draw the triangle!

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Law of Sines AAS/ASA (Example)

• Ex 1 Use the Law of Sines to solve the triangle
  round answers to two decimal places
• a) A 102.4, C 16.7, a 21.6
• b) A 55, B 42, c ¾

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SSA the Ambiguous Case

• Occurs when we know the length of two sides and
  the measure of the angle opposite one of two
  known sides
• e.g. a, b, A and b, c, C are SSA cases
• e.g. a, b, C and b, c, A are NOT (they are SAS
  cases)
• To solve the SSA case
• Use the Law of Sines to calculate the missing
  angle across from one of the known sides
• There are three possible cases

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SSA the Ambiguous Case (Continued)

• Let A be the missing angle across from one of the
  known sides and B be the measure of the known
  angle
• Case I sin A gt 1
• e.g. sin A 1.3511
• Recall the domain for the inverse sine -1 x
  1
• NO triangle exists
• Recall that the sine is positive in both Q I and
  Q II and how to use reference angles to obtain
  the angle in Q II
• Case II (180 A) B 180
• i.e. the measure of the possible second angle
  yields a second triangle with contradictory
  dimensions
• ONE triangle exists

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SSA the Ambiguous Case (Continued)

• Case III (180 A) B lt 180
• i.e. the measure of the possible second angle
  yields a second triangle with feasible dimensions
• TWO triangles exist

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SSA the Ambiguous Case (Example)

• Ex 2 Use the Law of Sines to solve for all
  solutions round to two decimal places
• a) A 54, a 7, b 10
• b) A 98, a 10, b 3
• c) C 27.83, c 347, b 425
• d) B 58, b 11.4, c 12.8

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Law of Sines Application (Example)

• Ex 3 A bridge is to be built across a small
  lake from a gazebo to a dock (see figure on page
  437). The bearing from the gazebo to the dock is
  S 41 W. From a tree 100 meters away from the
  gazebo, the bearings to the gazebo and dock are S
  74 E and S 28 E respectively. Find the length
  of the bridge.

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Summary

• After studying these slides, you should be able
  to
• Apply the Law of Sines in solving for the
  components of a triangle or in an application
  problem
• Additional Practice
• See the list of suggested problems for 6.1
• Next lesson
• Law of Cosines (Section 6.2)