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The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure). The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be (cos θ, sin θ), and then using the Pythagorean theorem to calculate the chord length: [2]
By extension, an angle subtended by a more complex geometric figure may be defined in terms of the figure's convex hull and its diameter; for example, the angle subtended by a tree as viewed in a camera (see angular size). [1] A subtended plane angle can also be defined for any two arbitrary isolated points and a vertex, as in two lines of ...
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 2α. The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that intercepts the same arc on the circle.
A circular segment (in green) is enclosed between a secant/chord (the dashed line) and the arc whose endpoints equal the chord's (the arc shown above the green area). In geometry , a circular segment or disk segment (symbol: ⌓ ) is a region of a disk [ 1 ] which is "cut off" from the rest of the disk by a straight line.
The crossing pattern of chords in a chord diagram may be described by a circle graph, the intersection graph of the chords: it has a vertex for each chord and an edge for each two chords that cross. [3] In knot theory, a chord diagram can be used to describe the sequence of crossings along the planar projection of a knot, with each point at ...
The angle between a chord and the tangent at one of its endpoints is equal to one half the angle subtended at the centre of the circle, on the opposite side of the chord (tangent chord angle). If the angle subtended by the chord at the centre is 90°, then ℓ = r √2, where ℓ is the length of the chord, and r is the radius of the circle.
The point of tangency can be constructed by the following steps: [4] [5] Reflect the bottom point of the painting across the line at eye-level. Construct the line segment between this reflected point and the top point of the painting. Draw the circle having this line segment as its diameter.
The fractional parts of chord lengths required great accuracy, and were given in sexagesimal notation in two columns in the table: The first column gives an integer multiple of 1 / 60 , in the range 0–59, the second an integer multiple of 1 / 60 2 = 1 / 3600 , also in the range 0–59.