technology in precalculus the ambiguous case of the law of sines & cosines lalu simcik cabrillo...
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Technology in Precalculus
The Ambiguous Case of the Law of Sines & Cosines
Lalu SimcikCabrillo 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])
3 22 5 6x x x
( 1)( 2)( 3)x x x
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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
3 25 12 14x x x
2( 7)( 2 2)x x x
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Gaussian Elimination
Vs. Creative Elimination / Substitution
And after two steps:
40
2 0
6 0
x y z
x y
x z
24
12
4
x
y
z
40
3 40
7 40
x y z
y z
y z
24
12
4
x
y
z
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Uniqueness Proof
Alternative determinant ‘zero check’
Checking answer at each re-writeCorrect algebra does not ‘move’ solutionUnique polynomial interpolation
40
3 40
20 80
x y z
y z
z
24
12
4
x
y
z
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Graphing Features
Two Dimension Example
Three Dimension Mesh Demo
3 2( ) 2 2 3 3 1y x x x x x y
2 2
2 2
sin( , )
x yf x y
x y
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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 WisconsinSolves 1000 x 1000 linear system on my low cost laptop in 3 seconds.No cost to studentsSoftware upgrades paid “by your tax dollars”Law of Sines & Cosines vs. more time for vectors, DeMoivre’s Thm, And geometric series. =
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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
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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:
ab
cbaC
2cos
2221
AbaAbc 222 sincos
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.
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).
Abccba cos2222
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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
2 2 2
2 2 2
2 cos
(2 cos ) ( ) 0
a b c bc A
c b A c b a
34.1493669177
23.9243088157
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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? B1+ B2 = ?
2 2 21cos
2
a c bB
ac
0 180oB
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Example CasesCase a b A roots
0 2 31 20.5o 2 complex
1 Rt 31sin20.5o
31 20.5o Double real positive
2 12 31 20.5o Two positive
1 Iso 31 31 20.5o One positive, one zero
1 32 31 20.5o 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 Sines Law 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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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
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Pro’s & Con’s
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 don’t know
www.cabrillo.edu/~lsimcik