Transcript Pythagoras & His Theorem

Pythagoras’ Theorem &
Trigonometry
Our Presenters &
Objectives
Boon Kah
• Proving the theorem
Beng Huat
• Applying the theorem
• The Chinese Proof
• Solving an Eye Trick
• Preservation of Area
– Applet Demo
• Pythagorean Triplets
• Class Activity –
Proving the theorem
using Similar
Triangles
Our Presenters &
Objectives
Lawrence Tang
• Fundamentals of
Trigonometry
• Appreciate the definition
of basic trigonometry
functions from a circle
• Apply the definition of
basic trigonometry
functions from a circle to
a square.
Keok Wen
• The derivation of the
double-angle formula
Getting to the “Point”
“Something
Interesting
”
Dad & Son
The Pythagoras
Theorem
The square described upon the hypotenuse
of a right-angled triangle is equal to the
sum of the squares described upon the
other two sides.
• Or algebraically speaking……
h2 = a2 + b2
h
b
a
The “Chinese” Proof
b
a
a
h
b
h
h
b
a
h
a
b
4(1/2 ab) + h2 = (a + b)2
2ab + h2 = a2 + 2ab + b2
h2 = a2 + b2
This proof appears in the Chou
pei suan ching, a text dated
anywhere from the time of Jesus
to a thousand years earlier
A Geometrical Proof
Most geometrical proofs revolve around the
concept of
“Preservation of Area”
Class Activity
How many similar triangles can you
see in the above triangle???
Use them to prove the Pythagoras’
Theorem again!
How to interest students
with Pythagoras Theorem
Challenge Their Minds
8 x 8 squares
= 64 squares
Challenge Their Minds
2
2
3
1
4
1
4
3
13 x 5 squares
= 65 squares ?
Using Pythagoras
Theorem
8
h1
h1
22
= (9+ 64)
3
= (73)
11
h2
h2
44
33
5
= (32 + 82)
= (22 + 52)
= (4+ 25)
= (29)
h1 + h2 = (73 + 29)
2
= 13.9292 units
Using Pythagoras
Theorem
2
5
4
3
h
3
1
13
h
= (52 + 132)
= (25+ 169)
= (194)
= 13.9283 units
Using Pythagoras
Theorem
h1
2
2
h
4
1
1
4
h2
3
h1 + h2 = 13.9292 units
h = 13.9283 units
h ≠ h1 + h2
3
Pythagorean Triplets
• 3 special integers
• Form the sides of right-angled
triangle
h
y
x
• Example: 3, 4 & 5
• Non-example: 5, 6 & √61
Trick for Teachers
• Give me an odd number, except 1
(small value)
• Form a Pythagorean Triplet
• Form a right-angled triangle where
sides are integers
Trick for Teachers
• Shortest side = n
• The other side = (n2 – 1)  2
• Hypotenuse = [(n2 – 1)  2] + 1
•
•
•
•
For e.g., if n = 2
Shortest side = 5
The other side = 12
Hypotenuse = 13
Trick for Teachers
• Why share this trick?
• Can use this to set questions on
Pythagoras Theorem with ease
Trigonometry
• Meaning of Sine,Cosine & Tangent
• Formal Definition of Sine,Cosine and
Tangent based on a unit circle
• Extension to the unit square
• Double Angle Formula
Meaning of “Sine”,
“Cosine” & “Tangent”
• Sine – From half chord to bosom/bay/curve
• Cosine – Co-Sine, sine of the complementary
angle
• Tangent – to touch
The Story of 3 Friends
Sine
Tangent
Cosine
Formal Definition of Sine and
Cosine
sin 
1

A (1,0)
cos 
Unit circle
Some Results from Definition
• Definition of tan :
sin 
cos 
• Pythagorean Identity:
sin2 + cos2 = 1
Common Definition of
Sine, Cosine & Tangent
What happens if
slant edge  1?
`
Opposite
length
sin 
By principal of similar triangles,
(Sin )/ 1 = opposite/slant length
(Cos )/1 = adjacent/slant length
cos 
adjacent length
(Sin ) /(Cos ) = opposite/adjacent
length
For visual students
Therefore for a given angle  in ANY right
angled triangle,
•
Adjacent Length
cos  =
Hypotenuse
•
Opposite Length
tan  = Adjacent Length
opposite
•
Opposite Length
sin  =
Hypotenuse

adjacent
Invasion by King Square!
Side
Tide
Coside
Extension to Non-Circular Functions
side 

A (1,0)
Unit Square
coside 
Some Results from definition
• Tide 
• BUT is
=
side  /coside 
side2  + coside2  = 1 ?
Pythagorean
Theorem for
Square Function

side 
coside 
For 0 <  < 45
coside  =1
side  = tan 
tide  = tan 
For 45 <  < 90
side  = 1
coside  =cot 
tide  = tan 
Corresponding Pythagorean Thm:
side2 + coside2  = sec2 
Corresponding Pythagorean Thm:
side2 + coside2  = cosec2 
Comparison of other theorems
Circular Function
Square Function
Complementary Thm
Supplementary Thm
Half Turn Thm
Opposites Thm
Comparison of Sine and Side
Functions
Comparison of Cosine and Coside
Functions
Comparison of Tan and Tide Functions
Further Extensions…
(0,1)
(0,1)
(1,0)
(1,0)
Diamond
Hexagon
References
• http://www.arcytech.org/java/pythagoras/history.html
• http://www-history.mcs.stand.ac.uk/history/Mathematicians/Pythagoras.html
• http://www.ies.co.jp/math/products/geo2/applets/pytha
2/pytha2.html
• The teaching of trigonometry in schools
London G Bell & Sons, Ltd
• Functions, Statistics & Trigonometry, Intergrated
Mathematics 2nd Edition,
University of Chicago School Math Project
Sine, Cosine &
Tangent
Opposite Length Adjacent Length Opposite Length
Slant length
Slant length
Adjacent length
1
For an angle ,
1
a
1
a
1
a
1
a
x
o
o
o
1
a
o
a
o
o
x(o)
x(a)
o defined
as sin 
= 2(o)/2
= o
= sin 
= 3(o)/3
= o
= sin 
= x(o)/x(1)
= o
= sin 
a defined
as sin 
= 2(a)/2
= a
= cos 
= 3(a)/3
= a
= cos 
= x(a)/x(1)
= a
= cos 
o/a defined
as tan 
= 2(o)/2(a)
= o/a
= tan 
= 3(o)/3(a)
= o/a
= tan 
= x(o)/x(a)
= o/a
= tan 
Return
Comparison of Complementary
Theorems
Square Function
Circular Function
For 0 <  < 90
For 0 <  < 45
sin(90 - ) = cos 
side(90 - ) = coside 
cos(90 - ) = sin 
coside(90 - ) = side 
tide(90 - ) = cotide 
tan(90 - ) = cot 

side (90-)
Unit Square
coside (90-)
Return
Comparison of functions of
(90 + )
Square Function
Circular Function
For 0 <  < 90
For 0 <  < 45
sin(90+ ) = cos 
side(90 + ) = coside 
cos(90+ ) = -sin 
coside(90 + ) = -side 
tan(90+ ) = -cot 
tide(90 + ) = -cotide 

side (90+)
Unit Square
coside (90+)
Return
Comparison of Supplement
Theorems
Square Function
Circular Function
For 0 <  < 90
For 0 <  < 45
sin(180 - ) = sin 
side(180 - ) = side 
cos(180 - ) = -cos
coside(180 - ) = -coside 
tan(180 - ) = -tan 
tide(180 - ) = -tide 
side (180-)

Unit Square
coside (180-)
Return
Comparison of ½ Turn Theorems
Square Function
Circular Function
For 0 <  < 90
For 0 <  < 45
sin(180 + ) = - sin 
side(180 + ) = - side 
cos(180 + ) = - cos
coside(180 + ) = - coside 
tide(180 + ) = tide 
tan(180 + ) = tan 
side (180+)

Unit Square
coside (180+)
Return
Comparison of Functions of (270 - )
Square Function
Circular Function
For 0 <  < 90
For 0 <  < 45
sin (270-) =-cos 
side(270 - ) = - coside 
cos(270-) = -side 
coside(270 - ) = - side 
tide(270 - ) = cotide 
tan (270-) = cot 
coside (270-)
side (270-)

Unit Square
Return
Comparison of Functions of (270 + )
Square Function
Circular Function
For 0 <  < 90
For 0 <  < 45
sin(270+ )= - cos 
side (270+ )= - coside 
cos(270+  ) = sin 
coside (270+ ) = side 
tide (270+ )= - cotide 
tan(270+) = - tan 
side (180-)

coside (270+)
Unit Square
Return
Comparison of Opposite Theorems
Circular Function
For 0 <  < 90
Square Function
For 0 <  < 45
sin(- ) = - sin 
side(- ) = - side 
cos(- ) = cos 
coside(- ) = coside 
tan(- ) = - tan 
tide(- ) = - tide 

side (-)
Unit Square
coside (-)
Return