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: 3 2
x 2 x 5x 6 By typing: roots([ 1 2 -5 -6]) ( x 1)( x 2)( x 3) Screen shot for polynomial roots: Fundamental Thm. of Algebra Students could soon handle with the help of long or synthetic division: x3 5 x 2 12 x 14 Via the real root x = 7 ( x 7)( x 2 2 x 2)
Gaussian Elimination Vs. Creative Elimination / Substitution x y z 40 x 2y 0 x 6z 0 x 24 y 12 z 4
And after two steps: x y z 40 3 y z 40 y 7 z 40 x 24 y 12 z 4 Uniqueness Proof Alternative determinant zero check x y z 40 3 y z 40
20 z 80 x 24 y 12 z 4 Checking answer at each re-write Correct algebra does not move solution Unique polynomial interpolation Graphing Features Two Dimension Example
y ( x) x 3 x 2 2 x 2 x 3 3 y 1 Three Dimension Mesh Demo f ( x, y ) sin
x2 y2 x2 y2 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, DeMoivres Thm, And geometric series. = 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 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: a2 b2 c2 C cos 2ab 1 Non-SAS case: c b cos A a 2 b 2 sin 2 A
. 2 2 2 a b c 2bc cos A 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). The textbook answer Encourage students to make an accurate sketch before solving each triangle With Octave
a 2 b 2 c 2 2bc cos A c 2 (2b cos A) c (b 2 a 2 ) 0 a=12 b=31 A=20.5 degrees roots([ 1 -2*b*cosd(A) b^2-a^2 ] ) Two real positive roots for c 34.1493669177 23.9243088157 Octave screen shot with a=12 Finding Angles Obtuse or Acute?
2 2 2 a c b 1 B cos
2 ac Find B or C first? 0 B 180o Results are not drawing-dependent Students might ask? B1+ B2 = ? Example Cases Case
a b 0 2 31 1 Rt
31sin20.5 2 12 A roots 20.5 o
2 complex 20.5 o Double real positive 31 20.5
o Two positive 1 Iso 31 31 20.5 o
One positive, one zero 1 31 20.5 o One positive, one negative
32 o 31 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 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 Pros & Cons Disadvantages: Learning Octave / Matlab PC / Mac access Round off error highly acute s Environment Smart rooms can help Improvement Metric
When lacking real data, talk about data Two SSA case on last exam Closing I dont know