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  • Geometry Arc Circle Area – PUTNAM Problem A2 1998 – Trigonometry Identity Proof

    Anonymous Christian
    Dec 2, 2018
    Math Olympiad, PUTNAM
    Comments Off on Geometry Arc Circle Area – PUTNAM Problem A2 1998 – Trigonometry Identity Proof

    Geometry Arc Circle Area

    I Found this Interesting Math Olympiad PUTNAM Problem and Its Solution (not mine) whilst browsing.

     

     

     

     

     

     

     

     

     

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    A Comment for edification

     

    A Very Clever Solution you have there.

     

    Comment: Why Multiply by 2?

     

    (i) Twice the First Picture in Addend

     

    Note that

     

    “First Picture in Addend” = (Area of Triangle) = (half of area of Top Left Rectangle)

     

    Thus doubling it,

     

    “Twice the First Picture in Addend” = Twice (half of area of Top Left Rectangle) = (Area of Top Left Rectangle)

     

    (ii) Twice the Second Picture in Addend

     

    Note that

     

    “Second Picture in Addend” = (Area of Triangle) = (half of area of Bottom Right Rectangle)

    Thus doubling it,

     

    “Twice the Second Picture in Addend” = Twice (half of area of Bottom Right Rectangle) = (Area of Bottom Right Rectangle)

     

    (iii) Twice the Third Picture in Addend

     

    Note that

     

    “Third Picture in Addend” = (Area of part-Arc)

     

    Thus doubling it,

     

    “Twice the Third Picture in Addend” = Twice (Area of part-Arc) = (Area of part-Arc COUNTED TWICE, one for its calculation in A and the other for its calculation in B.)

     

    Yes only this area is counted “twice” in the Overall Sum because it occurs in both “A and B respectively” .

     

    (iv) Total Shaded Area

     

    (Total Shaded Area) or (A + B.)

     

    = 2 x [ (half of area of Top Left Rectangle) + (half of area of Bottom Right Rectangle) + (Area of part-Arc)]

     

    = 2 x [ Area of Arc with angle “theta” & radius “1”]

     

    = 2 x [ (1/2) x r^2 x (theta) ], radius given as 1

     

    = 2 x [ (1/2) x 1^2 x (theta) ]

     

    = 2 x (1/2) x (theta)

     

    = (theta)

     

    = s

     

    Depending only on the arc length s, when radius is given as claimed.

     

    QED

     

    Note:

     

    Using (theta) in Radians,

     

    r = radius = 1

     

    (theta) = angle in Radians

     

    s = arc length

     

    Then,

     

    s = r (theta)

     

    s = (1) (theta)

     

    s = (theta) which was substituted in the last line above.

     

     

    P/S: A Visual Geometrical Proof  for the infamous elementary trigonometric identities

     

     

     

     

     

     

     

     

    How to understand the proof above?

    It’s difficult to explain here. Here are some useful pointers:

    Example: Look at shaded triangle in Second Picture above.

     

    Sin (a + b)

     

    = (Vertical dotted line)/ (Hypotenuse)

     

    = (Cos a Sin b + Sin a Cos b) / 1

     

    = (Cos a Sin b + Sin a Cos b)

     

    Can you see it?

     

    Note:

     

    (Vertical dotted line) = (Cos a Sin b + Sin a Cos b)

     

    By diagram construction.

     

    Example: In the right triangle with angle “a” in the Top Right Corner,

     

    It has (hypotenuse) = Sin b

     

    and an ‘angle a’.

     

    So, its

     

    Sin a = (‘horizontal length’)/(hypotenuse)

     

    Sin a = (‘horizontal length’)/(Sin b)

     

    So, rewriting this we obtain:

     

    (‘horizontal length’) = (Sin a) x (Sin b) as shown in Pic etc.

     

anonymous-christian

Jonathan Ramachandran is founder at AnonymousChristian.org which serves as a Christian Blog discussing the Greater Hope of Biblical Christian in Christ Alone. He is also an IMO (International Math Coach) and a featuring Guitarist for many bands.

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