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