Technology in Precalculus The Ambiguous Case of the Law of

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Technology in Precalculus

• The Ambiguous Case of the Law of Sines Cosines
• Lalu Simcik
• Cabrillo College

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Simplify Expand Resources

• What if, on day one of precalculus, students
  could factor polynomials like
• By typing roots( 1 2 -5 -6)

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Screen shot for polynomial roots
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Fundamental Thm. of Algebra

• Students could soon handle with the help of long
  or synthetic division
• Via the real root x 7

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Gaussian Elimination

• Vs. Creative Elimination / Substitution
• And after two steps

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Uniqueness Proof

• Alternative determinant zero check
• Checking answer at each re-write
• Correct algebra does not move solution
• Unique polynomial interpolation

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Graphing Features

• Two Dimension Example
• Three Dimension Mesh Demo

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Screen shot for 2-D plotting
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Screen shot for 3-D Mesh
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Octave is Matlab

• NSF with Univ. of Wisconsin
• Solves 1000 x 1000 linear system on my low cost
  laptop in 3 seconds.
• No cost to students
• Software upgrades paid by your tax dollars
• Law of Sines Cosines vs. more time for vectors,
  DeMoivres Thm, And geometric series.

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Background Oblique Triangles

• Third Century BC Euclid
• 15th Century Al-Kashi generalized in spherical
  trigonometry
• Popularized by Francois Viete, as is since the
  19th century.
• Wikipedia summarizes the method proposed here

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From Wikipedia

• Applications of the law of cosines unknown side
  and unknown angle.
• The third side of a triangle if one knows two
  sides and the angle between them

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Two Sides more known

• The angles of a triangle if one knows the three
  sides SSS
• Non-SAS case

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.

• The formula shown is the result of solving for c
  in the quadratic equation
• c2  - (2b cos A) c    (b2 - a2)
  0
• This equation can have 2, 1, or 0 positive
  solutions corresponding to the number of possible
  triangles given the data. It will have two
  positive solutions if
• b sin(A) lt a lt b
• only one positive solution if a gt b or
• a b sin(A), and no solution if a lt
  b sin(A).

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The textbook answer

• Encourage students to make an accurate sketch
  before solving each triangle

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With Octave

• a12 b31 A20.5 degrees
• roots( 1 -2bcosd(A) b2-a2 )
• Two real positive roots for c

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Octave screen shot with a12
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Finding Angles

• Obtuse or Acute? Find B or C first?
• Results are not drawing-dependent
• Students might ask? B1 B2 ?

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Example Cases
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Octave screen shot all cases
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Summary (for students)
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Pros Cons

• Advantages
• Accurate drawing not required
• After sketch is made at the end with available
  data, students can resolve supplementary /
  isosceles concepts more easily.
• Simplified structure for memorization
• Octave / Matlab skills resources

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Pros Cons

• Disadvantages
• Learning Octave / Matlab
• PC / Mac access
• Round off error highly acute ?s

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Environment

• Smart rooms can help

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Improvement Metric

• When lacking real data, talk about data
• Two ? SSA case on last exam

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Closing

• I dont know
• www.cabrillo.edu/lsimcik