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

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Presentation on theme: "Technology in Precalculus The Ambiguous Case of the Law of Sines & Cosines Lalu Simcik Cabrillo College."— Presentation transcript:

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([ ])

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, DeMoivres Th m, And geometric series. =

11 Background: Oblique Triangles Third Century BC: Euclid 15 th Century: Al-Kashi generalized in spherical trigonometry Popularized by Francois Viete, as is since the 19 th 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 equationquadratic equation c 2 (2b cos A) c + (b 2 a 2) = 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? B 1 + B 2 = ?

19 Example Cases CaseabAroots o 2 complex 1 Rt 31sin20.5 o o Double real positive o Two positive 1 Iso o One positive, one zero o One positive, one negative

20 Octave screen shot – all cases

21 Summary (for students) Two Angles plus more Two Sides plus more Law of SinesLaw of Cosines Unique solution No quadratic – no problem No acute / obtuse issue Only positive real roots create real triangles Find second angle with the Law of Cosines – naturally! Make drawings at the end when the triangle is resolved

22 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

23 Pros & Cons 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 dont know

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