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Technology in Precalculus

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Technology in Precalculus

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    1. Technology in Precalculus The Ambiguous Case of the Law of Sines & Cosines Lalu Simcik Cabrillo College

    2. Simplify & Expand Resources What if, on day one of precalculus, students could factor polynomials like: By typing: roots([ 1 2 -5 -6])

    3. Screen shot for polynomial roots:

    4. Fundamental Thm. of Algebra Students could soon handle with the help of long or synthetic division: Via the real root x = 7

    5. Gaussian Elimination Vs. Creative Elimination / Substitution And after two steps:

    6. Uniqueness Proof Alternative determinant ‘zero check’ Checking answer at each re-write Correct algebra does not ‘move’ solution Unique polynomial interpolation

    7. Graphing Features Two Dimension Example Three Dimension Mesh Demo

    8. Screen shot for 2-D plotting:

    9. Screen shot for 3-D Mesh:

    10. 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, DeMoivre’s Thm, And geometric series. =

    11. 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

    12. 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:

    13. Two Sides “+” more known: The angles of a triangle if one knows the three sides SSS: Non-SAS case:

    14. . 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) < a < b only one positive solution if a > b or a = b sin(A), and no solution if a < b sin(A).

    15. The textbook answer “Encourage students to make an accurate sketch before solving each triangle”

    16. With Octave a=12 b=31 A=20.5 degrees roots([ 1 -2*b*cosd(A) b^2-a^2 ] ) Two real positive roots for c

    17. Octave screen shot with a=12

    18. Finding Angles Obtuse or Acute? Find B or C first? Results are not drawing-dependent Students might ask? B1+ B2 = ?

    19. Example Cases

    20. Octave screen shot – all cases

    21. Summary (for students)

    22. Pro’s & Con’s 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

    23. Pro’s & Con’s Disadvantages: Learning Octave / Matlab PC / Mac access Round off error – highly acute ?’s

    24. Environment Smart rooms can help

    25. Improvement Metric When lacking real data, talk about data Two ? SSA case on last exam

    26. Closing I don’t know www.cabrillo.edu/~lsimcik

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