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