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
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Planning for Senior Year . And Beyond. Good Morning. We are here to share an important resource with you and let you know about some opportunities coming up that can help you with planning for your senior year.
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