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More generally, an angle subtended by an arc of a curve is the angle subtended by the corresponding chord of the arc. For example, a circular arc subtends the central angle formed by the two radii through the arc endpoints. If an angle is subtended by a straight or curved segment, the segment is said to subtend the angle.
Equal chords are subtended by equal angles from the center of the circle. A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle. If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem).
For fixed points A and B, the set of points M in the plane for which the angle ∠AMB is equal to α is an arc of a circle. The measure of ∠ AOB , where O is the center of the circle, is 2 α . The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2 θ that intercepts the same arc on the circle.
[1] [3] The half-chords were called ardha-jyās or jyā-ardhas. These terms were again shortened to jyā by omitting the qualifier ardha which meant "half of". The Sanskrit word koṭi has the meaning of "point, cusp", and specifically "the curved end of a bow". In trigonometry, it came to denote "the complement of an arc to 90°".
For tiny arcs, the chord is to the arc angle in degrees as π is to 3, or more precisely, the ratio can be made as close as desired to π / 3 ≈ 1.047 197 55 by making θ small enough. Thus, for the arc of 1 / 2 °, the chord length is slightly more than the arc angle in degrees. As the arc increases, the ratio of the chord to ...
The names exsecant, versine, chord, etc. can also be applied to line segments related to a circular arc. [2] The length of each segment is the radius times the corresponding trigonometric function of the angle. The external secant function (abbreviated exsecant, symbolized exsec) is a trigonometric function defined in terms of the secant function:
Angle AOB is a central angle. A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians). [1]
Since rail routes have very large radii, they are laid out in chords, as the difference to the arc is inconsequential; this made work easier before electronic calculators became available. The 100 feet (30.48 m) is called a station, used to define length along a road or other alignment, annotated as stations plus feet 1+00, 2+00, etc. Metric ...