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Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, the angle 2 π /5 radians (72° = 360°/5) can be trisected, but the angle of π /3 radians (60°) cannot be trisected. [8]
As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. Transformations are ways of shifting and scaling functions using different algebraic formulas.
In trigonometry, the gradian – also known as the gon (from Ancient Greek γωνία (gōnía) ' angle '), grad, or grade [1] – is a unit of measurement of an angle, defined as one-hundredth of the right angle; in other words, 100 gradians is equal to 90 degrees.
Trigonometry (from Ancient Greek τρίγωνον (trígōnon) ' triangle ' and μέτρον (métron) ' measure ') [1] is a branch of mathematics concerned with relationships between angles and side lengths of triangles.
The fourth angle of a Lambert quadrilateral is an obtuse angle in elliptic geometry. The summit angles of a Saccheri quadrilateral are obtuse in elliptic geometry. The sum of the measures of the angles of any triangle is greater than 180° if the geometry is elliptic. That is, the defect of a triangle is negative. [80]
The book advocates replacing the usual basic quantities of trigonometry, Euclidean distance and angle measure, by squared distance and the square of the sine of the angle, respectively. This is logically equivalent to the standard development (as the replacement quantities can be expressed in terms of the standard ones and vice versa).
Even with these restrictions, if the polar angle (inclination) is 0° or 180°—elevation is −90° or +90°—then the azimuth angle is arbitrary; and if r is zero, both azimuth and polar angles are arbitrary. To define the coordinates as unique, the user can assert the convention that (in these cases) the arbitrary coordinates are set to zero.
The first issue for geometers is what kind of geometry is adequate for a novel situation. Unlike in Euclidean geometry, the concept of an angle does not apply in projective geometry, because no measure of angles is invariant with respect to projective transformations, as is seen in perspective drawing from a changing perspective. One source for ...
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