MILTON HERRERA MATH PROJECT SPRING/2006 SINUSOID ORLANDO ALONSO Professor of Mathematics Math 115 LaGuardia Community College

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

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 ( BC). who tabulated the lengths of circle arcs with the lengths of the subtending chords.

History Ptolemy of Egypt (2nd century) expanded upon this work in his Almagest, and created a table of his results. Aryabhata (476–550), first defined the sine as the modern relationship between half an angle and half a chord.

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.

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.

A sinusoid is any function of time having the following form: X (t) = A sin (wt + φ) + 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 2πf (f in Hz).  P = **period or wavelength, the length of each cycle (2π/w).  T = time (sec).  f = frequency (Hz).  φ = 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.

Example Plots the sinusoid A sin (2πft+φ), for A = 10, f = 2.5, φ = π/4, and t Є [0,1].

All trig functions can be defined in terms of sine: A sin ( x / p + φ )  Sin[ θ ]  Cos[ θ ]:=Sin[ θ +π/2]  Tan[ θ ]:=Sin[ θ ]/Cos[ θ ]  Sec[ θ ]:=1/Cos[ θ ]  Csc[ θ ]:=1/Sin[ θ ]  Cot[ θ ]:=1/Tan[ θ ]

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).

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.

Wavy Surface A packaging form is modeled after the surface. Sin[x]*Sin[y].

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.