Technology in Precalculus

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

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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: By typing: roots([ 1 2 -5 -6]). Screen shot for polynomial roots:. - PowerPoint PPT Presentation

Transcript of Technology in Precalculus

Page 1: Technology in Precalculus

Technology in Precalculus

The Ambiguous Case of the Law of Sines & Cosines

Lalu SimcikCabrillo College

Page 2: Technology in Precalculus

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