MILTON HERRERA

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SINUSOID
ORLANDO ALONSO Professor of Mathematics Math
115 LaGuardia Community College

• MILTON HERRERA
• MATH PROJECT
• SPRING/2006

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SINUSOID

• Also called a sine wave.
• It is the graph of the sine function.
• It consists of a single frequency at constant
  amplitude.

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History

• Began in early civilization as a very important
  measuring science.
• There is evidence that the Babylonians first used
  it, it is written on a Babylonian cuneiform
  tablet.

• The earliest use of sine appears in the Sulba
  Sutras written in ancient India from the 8th
  century BC to the 6th century BC.
• Later studied by Hipparchus of Nicaea (180-125
  BC). who tabulated the lengths of circle arcs
  with the lengths of the subtending chords.

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History

• Ptolemy of Egypt (2nd century) expanded upon this
  work in his Almagest, and created a table of his
  results.
• Aryabhata (476550), first defined the sine as
  the modern relationship between half an angle and
  half a chord.

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History

• The Indian works were later translated and
  expanded by Muslim mathematicians.
• From them the knowledge probably passed to the
  Greeks.
• Pythagoras.
• It captures the idea of a wave, a fundamental
  concept in physics.
• Joseph Furier studied the mathematical theory of
  heat conduction.

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Description

• Sine is sometimes called circular function
  because the essential feature of the sine
  function can be thought of as a point moving
  around a circle in a uniform way, and the value
  of sine being the height of the point.

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A sinusoid is any function of time having the
following form X (t) A sin (wt f) C

• Where all variables are real numbers, and
• A the amplitude, the height of each peak above
  the baseline (nonnegative)
• w the angular frequency (rad/sec) given by 2pf
  (f in Hz).
• P period or wavelength, the length of each
  cycle (2p/w).

• T time (sec).
• f frequency (Hz).
• f the phase shift, the horizontal offset of the
  basepoint where the curve crosses the baseline
  as it ascends. Also call initial phase (radians).
• C the vertical offset, height of the baseline.
  

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Example

• Plots the sinusoid A sin (2pftf), for A 10 , f
  2.5, f p/4, and t ? 0,1.

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All trig functions can be defined in terms of
sine A sin ( x / p f )

• Sin?
• Cos?Sin?p/2
• Tan?Sin?/Cos?
• Sec?1/Cos?
• Csc?1/Sin?
• Cot?1/Tan?

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APLICATIONS OF THE SINOSOID

• Development of cut cylinder Sinusoid is the
  development of an obliquely cut right circular
  cylinder. (The edge of the cylinder rolled out is
  a sinusoid).

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The cylinder cut method is used to generate these
parts.

• This example shows a piston an important
  component an engine.
• A gear is a toothed wheel designed to transmit
  torque. This is the principle of the automobile
  transmission.

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Wavy Surface

• A packaging form is modeled after the surface.
• SinxSiny.

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Helix Projection

• Sinusoid is the orthogonal projection of the
  space curve helix.
• A helicoid is a surface formed as the trace of a
  rotating a line along an axis.