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

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

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Screen shot for polynomial roots:

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

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

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

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Graphing Features Two Dimension Example Three Dimension Mesh Demo

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Screen shot for 2-D plotting:

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Screen shot for 3-D Mesh:

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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. =

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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

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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:

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

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. 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).

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

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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

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Octave screen shot with a=12

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

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Example Cases CaseabAroots 0 23120.5 o 2 complex 1 Rt 31sin20.5 o 3120.5 o Double real positive 2 123120.5 o Two positive 1 Iso 31 20.5 o One positive, one zero 1 323120.5 o One positive, one negative

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Octave screen shot – all cases

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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

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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

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Pros & Cons Disadvantages: Learning Octave / Matlab PC / Mac access Round off error – highly acute s

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Environment Smart rooms can help

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Improvement Metric When lacking real data, talk about data Two SSA case on last exam

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Closing I dont know www.cabrillo.edu/~lsimcik

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In section 9.2 we mentioned that by the SAS condition for congruence, a triangle is uniquely determined if the lengths of two sides and the measure of.

In section 9.2 we mentioned that by the SAS condition for congruence, a triangle is uniquely determined if the lengths of two sides and the measure of.

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