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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.

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## Pythagoras’ Theorem & Trigonometry

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**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

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