540 likes | 1.02k Views
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. Beng Huat. Applying the theorem Solving an Eye Trick Pythagorean Triplets.
E N D
Our Presenters & Objectives • 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
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 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 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
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!
8 x 8 squares = 64 squares Challenge Their Minds
2 2 3 1 4 1 4 3 Challenge Their Minds 13 x 5 squares = 65 squares ?
8 h1 2 2 3 1 1 h2 4 4 3 3 5 2 Using Pythagoras Theorem 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 Using Pythagoras Theorem 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 Using Pythagoras Theorem h ≠h1 + h2
h y x Pythagorean Triplets • 3 special integers • Form the sides of right-angled triangle • 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
Sine Tangent Cosine The Story of 3 Friends
sin A (1,0) cos Formal Definition of Sine and Cosine 1 Unit circle
Some Results from Definition • Definition of tan : sin cos • Pythagorean Identity: • sin2 + cos2 = 1
` slant length Opposite length 1 sin cos adjacent length Common Definition of Sine, Cosine & Tangent 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 Invasion by King Square!
side coside Extension to Non-Circular Functions A (1,0) Unit Square
Some Results from definition • Tide = side /coside • BUT is side2 + coside2 = 1 ?
side Corresponding Pythagorean Thm: side2+ coside2 = sec2 coside Corresponding Pythagorean Thm: side2+ coside2 = cosec2 Pythagorean Theorem for Square Function For 0 < < 45 coside =1 side = tan tide = tan For 45 < < 90 side = 1 coside =cot tide = tan
Comparison of other theorems Circular FunctionSquare Function Complementary Thm Supplementary Thm Half Turn Thm Opposites Thm AGREES !!
Further Extensions… (0,1) (0,1) (1,0) (1,0) Hexagon Diamond
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
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-) 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 Return
side (90+) Unit Square coside (90+) 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 Return
side (180-) Unit Square coside (180-) Comparison of Supplement 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
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