1 the meaning of these terms is apparent if one looks

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[1] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle, shown at right. For a vertical chord AB of the unit circle, the sine of the angle θ (half the subtended angle) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos( θ ) = OC and versin( θ ) = CD is the radius OD = 1. Illustrated this way, the sine is vertical ( rectus ) while the versine is flipped on its side ( versus ); both are distances from C to the circle. This figure also illustrates the reason why the versine was sometimes called the sagitta , Latin for arrow, from the Arabic usage sahem of the same meaning. [5] If the arc ADB is viewed as a "bow" and the chord AB as its "string", then the versine CD is clearly the "arrow shaft". In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal", sagitta is also an obsolete synonym for the abscissa (the horizontal axis of a graph). [5] One period (0 < θ < π /2) of a versine or, more commonly, a haversine waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function, because it smoothly (continuous in value and slope) "turns on" from zero to one (for haversine) and back to zero. In these applications, it is given yet another name: raised-cosine filter or Hann function. "Versines" of arbitrary curves and chords The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length L goes to zero, the ratio 8 v / L 2 goes to the instantaneous curvature. This usage is especially common in rail transport, where it describes measurements of the straightness of the rail tracks [6] and it is the basis of the Hallade method for rail surveying. The term sagitta (often abbreviated sag ) is used similarly in optics, for describing the surfaces of lenses and mirrors.
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  • Spring '12
  • MUNK
  • Trigonometry, Haversine formula, free encyclopedia, Versine, Exsecant

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