Chapter 6.1 Law of Sines

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Chapter 6
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Chapter 6.1 Law of Sines

• In Chapter 4 you looked at techniques for solving
  right triangles.
• In this section and the next section you will
  solve oblique triangles.
• Oblique triangles are triangles that have no
  right angle.
• As standard notation, the angles of a triangle
  are labeled A, B,and C and their opposite angles
  are labeled a,b,and c.

C
Note Angle A is between b and c, B is between a
and c, C is between a and b.
a
b
B
A
c
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Chapter 6.1 Law of Sines

• To solve an oblique triangle, you need to know
  the measure of at least one side and the measures
  of any two other parts of the triangle- two
  sides, two angles, or one angle and one side.
• This breaks down into four cases
• Two angles and any side (AAS or ASA)
• Two sides and an angle opposite one of them (SSA)
• Three sides (SSS)
• Two sides and their included angle (SAS)

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Derive Law of Sines
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Chapter 6.1 Law of Sines1
NOTE The Law of Sines can also be written in the
reciprocal form
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Example 1 GIVEN Two Angles and One Side-AAS

• C102.3, B28.7, and b27.4 feet
• Find the remaining angle
• and sides
• A180 - B - C
• 180 -
• By the Law of Sines you
• have

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Try 3 page 398

• Use Law of Sines to Solve the triangle

GIVEN A10, a4.5, B60
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Example2 GIVEN Two Angles and One Side-ASA

• A pole tilts toward the sun at an 8 angle from
  the vertical, and it casts a 22 foot shadow. The
  angle of elevation from the tip of the shadow to
  the top of the pole is 43. How tall is the pole?

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Try 27 page 398

• Use Law of Sines to Solve the triangle

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The Ambiguous Case (SSA)

• If two sides and one opposite angle are given,
  three possible situations can occur (1) no such
  triangle exists, (2) one such triangle
• exists, or (3) two distinct triangles satisfy
  the conditions

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Example 3 Single-Solution Case (SSA)

• One solution a gt b

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Try 15 page 398

• Use Law of Sines to Solve the triangle

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Example 4 No-Solution Case SSA

• Show that there is no triangle for which a 15,
  b 25, and A 85º

No solution a lt h
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Try 17 page 398

• Use Law of Sines to Solve the triangle

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Example 5 Two Solution Case

• Find two triangles for which a 12 meters, b
  31 meters, and A 20.5º

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Try 19 page 398

• Use Law of Sines to Solve the triangle

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• Area of a Oblique Triangle

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Example 6 Area of an Oblique Triangle

• Find the area of a triangular lot having two
  sides of lengths 90 meters and 52 meters and an
  included angle of 102

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Try 21 page 398

• Find the area of the triangle

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Example 7 An Application of the Law of Sines

• The course for a boat race starts at point A and
  proceeds in the direction S 52 W to point B,
  then in the direction S 40 E to point C, and
  finally back to A. Point C lies 8 kilometers
  directly south of point A. Approximate the total
  distance of the race course.

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Try 29 page 398

• Use Law of Sines to Solve the triangle

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THEEND