The Pythagorean Theorem - Educational Outreach

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This one characteristic alone suggests that (at least intheory) trigonometric functions can be used to model thebehavior of any natural or contrived phenomena having anoscillating or repeating action over time such as theinduced ground-wave motion of earthquakes or the inducedatmospheric-wave motion due to various types of music orvoice transmission.C( x,y)1yFigure 3.16: Trigonometry via General Right TriangleThe six trigonometric functions also can be definedusing a general right triangle. Let ABC be an arbitraryright triangle with angle as shown on the previous pagein Figure 3.16. The triangle ABC is similar totriangle ( 0,0),(1,0),( x,y)guarantees:1]3]5]( 0,0) x (1,0 )ABy BCACsin( ) 2] csc()1 AC 1y BCx ABACcos( ) 4] sec()1 AC 1x ABy BCx ABtan( ) 6] cot( ) x ABy AC. This right-triangle similarity120
The following four relationships are immediatelyevident from the above six definitions. These fourrelationships will comprise our primitive trigonometricidentities.1]3]1csc( ) 2]sin( )sin( )tan( ) 4]cos( )1sec( )cos( )cos( )cot( )sin( )Since the four trigonometric functions csc( ) , sec( ) ,tan( ) , and cot( ) reduce to nothing more thancombinations of sines and cosines, one only needs to knowthe value of sin( ) and )cos( for a given in order toevaluate these remaining four functions. In practice,trigonometry is accomplished by table look up. Tables forthe six trigonometric function values versus are set up00for angles from 0 to 90 in increments of 10 ' or less.Various trigonometric relationships due to embeddedsymmetry in the unit circle were used to producetrigonometric functional values corresponding to angles00ranging from 90 to 360 . Prior to 1970, these values weremanually created using a book of mathematical tables.Nowadays, the painstaking operation of table lookup istransparent to the user via the instantaneous modernelectronic calculator.Our second pillar is a group of three identities,collectively called the Pythagorean identities. ThePythagorean identities are a direct consequence of thePythagorean Theorem. From the Pythagorean Theorem andthe definitions of sin(t)and cos(t ) , we immediately seethat22sin ( t ) cos ( t) 1 .121
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The PythagoreanTheoremCrown Jewel o
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The Pythagorean TheoremCrown Jewel
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Table of ContentsList of Tables and
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List of Tables and FiguresTablesNum
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Figures…continuedNumber and Title
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List of Proofs and DevelopmentsSect
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PrefaceThe Pythagorean Theorem has
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1 : 2x2x2****22 13x 7 13x 7 0 : a
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1) Consider the Squares“If it was
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s090045 0ss45450450090Figure 1.3: O
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Notice that the right-triangle prop
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The middle square (minus the donut
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Our proof in Chapter 1 has been by
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2) Four Thousand Years of Discovery
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The proof Pythagoras is thought to
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cabcFigure 2.4: Annotated Square wi
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Figure 2.6 is the diagram for a sec
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1x:b2:Area(CBA)3:Area( ABC)Area(ABC
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Euclid’s proof of the Pythagorean
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First we establish that the two tri
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1 2y b b: y b c c2:A3:A4:Aunshaded
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XB'A090 CBFigure 2.12: Euclid’s C
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2.3) Liu Hui Packs the SquaresLiu H
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One of my favorites was Boxel, a ga
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2.4) Kurrah Transforms the Bride’
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Note: as is the occasional custom i
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2.5) Bhaskara Unleashes the Power o
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2.6) Leonardo da Vinci’s Magnific
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In Figure 2.24, we enlarge Step 5 a
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But, this is precisely the nature o
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2.8) Henry Perigal’s TombstoneHen
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In the century following Perigal, b
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Returning to Henry Perigal, Figure
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Figure 2.32 is President Garfield
Page 71 and 72: Even though the Cartesian coordinat
Page 73 and 74: B 2← γ →α y βx C-xC 2Figure
Page 75 and 76: Being polynomial in form, the funct
Page 77 and 78: Thus, the critical point ( C 2 ,0)i
Page 79 and 80: Sincexcpwas chosen on an arbitrary
Page 81 and 82: ∂F/∂x = ∂F/∂y = 0x(C - x) 2
Page 83 and 84: Define a new function21 A2 A3 2xcp
Page 85 and 86: 2.11) Shear, Shape, and AreaOur las
Page 87 and 88: Step 1: Cut the big square into two
Page 89 and 90: In Table 2.3, we briefly summarize
Page 91 and 92: We can formally state this similari
Page 93 and 94: What we will do in Section 3.2 is p
Page 95 and 96: There are three Pythagorean-triple
Page 97 and 98: Thus, if we multiply any given prim
Page 99 and 100: The last equality shows that the in
Page 101 and 102: A Pythagorean Quartet is a set of f
Page 103 and 104: 3.5) Pythagoras and the Three Means
Page 105 and 106: yxy MHxab2ab2 y x FC y ab2ab 2ab y
Page 107 and 108: Hero was the founder of the Higher
Page 109 and 110: 5 : Solve for area using the formul
Page 111 and 112: 4 : Construct the parallelogram wit
Page 113 and 114: We already have introduced Thabit i
Page 115 and 116: Some historians believe Stewart’s
Page 117 and 118: In pre-computer days, both all four
Page 119 and 120: One example of a periodic process i
Page 121: The six trigonometric functions are
Page 125 and 126: As with any set of identities, the
Page 127 and 128: Figure 3.17 can be used as a jumpin
Page 129 and 130: The remaining two pillars are the L
Page 131 and 132: a2Law of Cosines c2 b2 2bccos()The
Page 133 and 134: An example of a Diophantine equatio
Page 136 and 137: However, Euler’s Conjecture did n
Page 138 and 139: 4) Pearls of Fun and WonderPearls o
Page 140 and 141: We have achieved five-digit accurac
Page 142 and 143: 140212213116)74370116)(74370(370)74
Page 144 and 145: X X X X O X X O X XO X X O O OO X O
Page 146 and 147: 4.3) Earth, Moon, Sun, and StarsIn
Page 148 and 149: The four-step solution follows. All
Page 150 and 151: Figure 4.9 depicts Eratosthenes’
Page 152 and 153: ---High above the earth, we see the
Page 154 and 155: The commonly accepted value is abou
Page 156 and 157: 1BC : tan( BAC)BABC BAtan(BAC)0BC
Page 158 and 159: BD ACtan( 21 )Even for the closest
Page 160 and 161: 3w 2h s w 5 s w 0.38196s&5 1s
Page 162 and 163: Figure 4.16: Triangular Phi---The h
Page 164 and 165: What remains to be done is to devel
Page 166 and 167: 3 : h32 4 h222 2 24C 2 h33162 2 2
Page 168 and 169: Epilogue: The Crown and the Jewels
Page 170 and 171: Not only has it endured, but the Py
Page 172 and 173:
A] Greek AlphabetGREEK LETTERUpper
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C] Geometric FoundationsThe Paralle
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8. Congruent Triangles: Two triangl
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3. Rectangle: A bh : P 2b 2h, b &
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Recreational Mathematics11. Pickove
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GGolden RatioDefined and algebraica
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Puzzles (cont)Pythagorean Magic Squ
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Pythagorean Theorem ProofsAlgebraic
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Trigonometry (cont)Eratosthenes mea
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The Pythagorean Theorem - Educational Outreach

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