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Pythagoras’ Theorem & Trigonometry

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Pythagoras’ Theorem & Trigonometry

- Proving the theorem
- The Chinese Proof
- Preservation of Area – Applet Demo
- Class Activity – Proving the theorem using Similar Triangles

Boon Kah

Beng Huat

- Applying the theorem
- Solving an Eye Trick
- Pythagorean Triplets

- 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

Keok Wen

- The derivation of the double-angle formula

“Something Interesting”

Dad & Son

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

b

a

b

h

a

h

4(1/2 ab) + h2 = (a + b)2

2ab + h2 = a2 + 2ab + b2

h

h2 = a2 + b2

a

h

b

This proof appears in the Chou pei suan ching, a text dated anywhere from the time of Jesus to a thousand years earlier

a

b

Most geometrical proofs revolve around the concept of

“Preservation of Area”

How many similar triangles can you see in the above triangle???

Use them to prove the Pythagoras’ Theorem again!

8 x 8 squares

= 64 squares

2

2

3

1

4

1

4

3

13 x 5 squares

= 65 squares ?

8

h1

2

2

3

1

1

h2

4

4

3

3

5

2

h1= (32 + 82)

= (9+ 64)

= (73)

h2= (22 + 52)

= (4+ 25)

= (29)

h1 + h2= (73 + 29)

= 13.9292 units

h

5

2

3

4

1

13

3

h= (52 + 132)

= (25+ 169)

= (194)

= 13.9283 units

h

h1

2

2

3

1

4

1

h2

4

3

h1 + h2 = 13.9292 units

h = 13.9283 units

h ≠h1 + h2

h

y

x

- 3 special integers
- Form the sides of right-angled triangle

- Example: 3, 4 & 5
- Non-example: 5, 6 & √61

- Give me an odd number, except 1 (small value)
- Form a Pythagorean Triplet
- Form a right-angled triangle where sides are integers

- 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

- Why share this trick?
- Can use this to set questions on Pythagoras Theorem with ease

- 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

- Sine – From half chord to bosom/bay/curve

- Cosine – Co-Sine, sine of the complementary
- angle

- Tangent – to touch

Sine

Tangent

Cosine

sin

A (1,0)

cos

1

Unit circle

- Definition of tan :
sin

cos

- Pythagorean Identity:
- sin2 + cos2 = 1

`

slant length

Opposite

length

1

sin

cos

adjacent length

What happens if slant edge 1?

By principal of similar triangles,

(Sin )/ 1 = opposite/slant length

(Cos )/1 = adjacent/slant length

(Sin ) /(Cos ) = opposite/adjacent length

For visual students

hypotenuse

opposite

adjacent

Therefore for a given angle in ANY right

angled triangle,

Opposite Length

- sin = Hypotenuse
Adjacent Length

- cos = Hypotenuse
Opposite Length

- tan = Adjacent Length

Side

Tide

Coside

side

coside

A (1,0)

Unit Square

- Tide = side /coside
- BUT is side2 + coside2 = 1 ?

side

Corresponding Pythagorean Thm:

side2+ coside2 = sec2

coside

Corresponding Pythagorean Thm:

side2+ coside2 = cosec2

For 0 < < 45

coside =1

side = tan

tide = tan

For 45 < < 90

side = 1

coside =cot

tide = tan

Circular FunctionSquare Function

Complementary Thm

Supplementary Thm

Half Turn Thm

Opposites Thm

AGREES !!

(0,1)

(0,1)

(1,0)

(1,0)

Hexagon

Diamond

- 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

1

o

a

1

= 2(o)/2

= o

= sin

o

= 2(o)/2(a)

= o/a

= tan

1

= 2(a)/2

= a

= cos

o

a

a

1

= 3(o)/3

= o

= sin

o

= 3(a)/3

= a

= cos

= 3(o)/3(a)

= o/a

= tan

1

o

1

o

a

a

a

= x(o)/x(1)

= o

= sin

= x(a)/x(1)

= a

= cos

= x(o)/x(a)

= o/a

= tan

x

x(o)

x(a)

Sine, Cosine & Tangent

Opposite Length

Slant length

Adjacent Length

Slant length

Opposite Length

Adjacent length

o defined

as sin

a defined

as sin

o/a defined

as tan

For an angle ,

Return

side (90-)

Unit Square

coside (90-)

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

Return

side (90+)

Unit Square

coside (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

Return

side (180-)

Unit Square

coside (180-)

Square Function

Circular Function

For 0 < < 90

For 0 < < 45

side(180 - ) = side

sin(180 - ) = sin

coside(180 - ) = -coside

cos(180 - ) = -cos

tide(180 - ) = -tide

tan(180 - ) = -tan

Return

side (180+)

Unit Square

coside (180+)

Comparison of ½ Turn Theorems

Square Function

Circular Function

For 0 < < 90

For 0 < < 45

side(180 + ) = - side

sin(180 + ) = - sin

coside(180 + ) = - coside

cos(180 + ) = - cos

tide(180 + ) = tide

tan(180 + ) = tan

Return

coside (270-)

side (270-)

Unit Square

Comparison of Functions of (270 - )

Square Function

Circular Function

For 0 < < 90

For 0 < < 45

side(270 - ) = - coside

sin (270-) =-cos

cos(270-) = -side

coside(270 - ) = - side

tide(270 - ) = cotide

tan (270-) = cot

Return

side (180-)

Unit Square

coside (270+)

Square Function

Circular Function

For 0 < < 90

For 0 < < 45

Comparison of Functions of (270 + )

side (270+ )= - coside

sin(270+ )= - cos

coside (270+ ) = side

cos(270+ ) = sin

tide (270+ )= - cotide

tan(270+) = - tan

Return

Square Function

Circular Function

For 0 < < 90

For 0 < < 45

Comparison of Opposite Theorems

side(- ) = - side

sin(- ) = - sin

cos(- ) = cos

coside(- ) = coside

tan(- ) = - tan

tide(- ) = - tide

side (-)

Unit Square

coside (-)

Return