Technology in Precalculus The Ambiguous Case of the

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

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

Screen shot for polynomial roots:

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

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

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

Graphing Features Two Dimension Example Three Dimension Mesh Demo

Screen shot for 2 -D plotting:

Screen shot for 3 -D Mesh:

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, De. Moivre’s Thm, And geometric series. =

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

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:

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

. The formula shown is the result of solving for c in the quadratic equation c 2 − (2 b 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).

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

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

Octave screen shot with a=12

Finding Angles Obtuse or Acute? Find B or C first? Results are not drawing-dependent Students might ask? B 1+ B 2 = ?

Example Cases Case a 0 2 1 Rt 31 sin 20. 5 2 12 1 Iso 31 1 32 b 31 o 31 31 A roots 20. 5 o 2 complex 20. 5 o Double real positive 20. 5 o Two positive 20. 5 o One positive, one zero 20. 5 o One positive, one negative

Octave screen shot – all cases

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

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