Technology in Precalculus

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