Pythagoras’ Theorem & Trigonometry
Our Presenters & Objectives
• Proving the theorem
• The Chinese Proof
• Preservation of Area – Applet Demo
• Class Activity – Proving the theorem using Similar Triangles
Boon Kah• Applying the theorem
• Solving an Eye Trick
• Pythagorean Triplets
Beng Huat
Our Presenters & Objectives
• 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.
Lawrence Tang• The derivation of the
double-angle formula
Keok Wen
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
b
a
h
The “Chinese” Proofab
a
a
a
b
b
b4(1/2 ab) + h2 = (a + b)2
2ab + h2 = a2 + 2ab + b2
hh
hh
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
12
34
1
23
4
13 x 5 squares
= 65 squares ?
Challenge Their Minds
Using Pythagoras Theorem
h1 = (32 + 82)
= (9+ 64)
= (73)
h2 = (22 + 52)
= (4+ 25)
= (29)
h1 + h2 = (73 + 29)
= 13.9292 units
12
34
2
8
3
5
12
34h2
h1
Using Pythagoras Theorem
3
h = (52 + 132)
= (25+ 169)
= (194)
= 13.9283 units
1
23
4h
13
5
Using Pythagoras Theorem
h = 13.9283 units
h1 + h2 = 13.9292 units
1
23
4h
h ≠ h1 + h2
12
34h2
h1
Pythagorean Triplets
• 3 special integers
• Form the sides of right-angled triangle
• Example: 3, 4 & 5
• Non-example: 5, 6 & √61
x
yh
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
Cosine
Tangent
A (1,0)
Formal Definition of Sine and Cosine
sin
cos
Unit circle
1
Some Results from Definition
• Definition of tan : sin cos
• Pythagorean Identity:
sin2 + cos2 = 1
`
By principal of similar triangles,
(Sin )/ 1 = opposite/slant length
(Cos )/1 = adjacent/slant length
(Sin ) /(Cos ) = opposite/adjacent length
What happens if slant edge 1?
1
cos
sin
Opposite length
adjacent length
slan
t len
gth
Common Definition of Sine, Cosine & Tangent
For visual students
Therefore for a given angle in ANY right angled triangle,
Opposite Length
• sin = Hypotenuse
Adjacent Length
• cos = Hypotenuse
Opposite Length
• tan = Adjacent Length
opposite
adjacent
hypoten
use
Invasion by King Square!
Side
Coside
Tide
Extension to Non-Circular Functions
Unit Square
A (1,0)
side
coside
Some Results from definition
• Tide = side /coside
• BUT is side2 + coside2 = 1 ?
Pythagorean Theorem for Square Function
For 0 < < 45coside =1side = tan tide = tan
Corresponding Pythagorean Thm:
side2 + coside2 = sec2
Corresponding Pythagorean Thm:
side2 + coside2 = cosec2
For 45 < < 90side = 1coside =cot tide = tan
side
coside
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…
(1,0)(1,0)
(0,1)(0,1)
Diamond Hexagon
References
• http://www.arcytech.org/java/pythagoras/history.html• http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Pythagoras
.html• http://www.ies.co.jp/math/products/geo2/applets/pytha2/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
Opposite Length Slant length
Adjacent Length Slant length
Opposite Length Adjacent length
o definedas sin
a definedas sin
o/a definedas tan
Sine, Cosine & Tangent
For an angle ,
1
a
1
ao
o = 2(o)/2 = o= sin
= 2(a)/2 = a= cos
= 2(o)/2(a)= o/a= tan
oo
o11
1
aa a
= 3(o)/3 = o= sin
= 3(a)/3 = a= cos
= 3(o)/3(a)= o/a= tan
x(a)
xx(o)
= x(o)/x(1) = o= sin
= x(a)/x(1) = a = cos
= x(o)/x(a)= o/a= tan
1o
a
Return
Comparison of Complementary TheoremsCircular Function
sin(90 - ) = cos
cos(90 - ) = sin
tan(90 - ) = cot
For 0 < < 90
side(90 - ) = coside
coside(90 - ) = side
tide(90 - ) = cotide
For 0 < < 45
Square Function
side (90-)
coside (90-)
Unit Square
Return
Comparison of functions of (90 + )
Circular Function
sin(90+ ) = cos
cos(90+ ) = -sin
tan(90+ ) = -cot
For 0 < < 90
side(90 + ) = coside
coside(90 + ) = -side
tide(90 + ) = -cotide
For 0 < < 45
Square Function
Return
side (90+)
coside (90+)
Unit Square
Comparison of Supplement Theorems
Circular Function
sin(180 - ) = sin
cos(180 - ) = -cos
tan(180 - ) = -tan
For 0 < < 90
side(180 - ) = side
coside(180 - ) = -coside
tide(180 - ) = -tide
For 0 < < 45
Square Function
side (180-)
coside (180-)
Unit Square
Return
Comparison of ½ Turn Theorems
Circular Function
sin(180 + ) = - sin
cos(180 + ) = - cos
tan(180 + ) = tan
For 0 < < 90
side(180 + ) = - side
coside(180 + ) = - coside
tide(180 + ) = tide
For 0 < < 45
Square Function
side (180+)
coside (180+)
Unit Square
Return
Comparison of Functions of (270 - )
Circular Function
sin (270-) =-cos
cos(270-) = -side
tan (270-) = cot
For 0 < < 90
side(270 - ) = - coside
coside(270 - ) = - side
tide(270 - ) = cotide
For 0 < < 45
Square Function
side (270-)
coside (270-)
Unit SquareReturn
Circular Function
sin(270+ )= - cos
cos(270+ ) = sin
tan(270+) = - tan
For 0 < < 90
side (270+ )= - coside
coside (270+ ) = side
tide (270+ )= - cotide
For 0 < < 45
Square Function
Comparison of Functions of (270 + )
side (180-)
coside (270+)
Unit Square
Return
Circular Function
sin(- ) = - sin
cos(- ) = cos
tan(- ) = - tan
For 0 < < 90
side(- ) = - side
coside(- ) = coside
tide(- ) = - tide
For 0 < < 45
Square Function
Comparison of Opposite Theorems
side (-)
coside (-)
Unit Square
Return