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Topic: Spherical trigonometry


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  Trigonometry - MSN Encarta
The two branches of trigonometry are plane trigonometry, which deals with figures lying wholly in a single plane, and spherical trigonometry, which deals with triangles that are sections of the surface of a sphere.
The earliest applications of trigonometry were in the fields of navigation, surveying, and astronomy, in which the main problem generally was to determine an inaccessible distance, such as the distance between the earth and the moon, or of a distance that could not be measured directly, such as the distance across a large lake.
Other applications of trigonometry are found in physics, chemistry, and almost all branches of engineering, particularly in the study of periodic phenomena, such as vibration studies of sound, a bridge, or a building, or the flow of alternating current.
encarta.msn.com /encyclopedia_761572350/Trigonometry.html   (1318 words)

  
 Trigonometry - MSN Encarta
Spherical trigonometry, which is used principally in navigation and astronomy, is concerned with spherical triangles, that is, figures that are arcs of great circles (see Navigation) on the surface of a sphere.
In spherical trigonometry, as well as in plane, three elements taken at random may not satisfy the conditions for a triangle, or they may satisfy the conditions for more than one.
Spherical trigonometry is of great importance in the theory of stereographic projection and in geodesy.
encarta.msn.com /encyclopedia_761572350_2/Trigonometry.html   (1498 words)

  
 spherical trigonometry - Article and Reference from OnPedia.com
Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles.
Note that this arc angle, measured in radians, and multiplied by the sphere's radius, is the arc length.
Hence, a spherical triangle is specified as usual by its corner angles and its sides, but the sides are not given by their length, but by their arc angle.
www.onpedia.com /encyclopedia/Spherical-trigonometry   (429 words)

  
 Trigonometry - LoveToKnow 1911
History Trigonometry, in its essential form of showing how to deduce the values of the angles and sides of a triangle when other angles and sides are given, is an invention of the Greeks.
A spherical triangle is the portion of the surface of a sphere bounded by three arcs of great circles of the sphere.
Analytical trigonometry is that branch of mathematical analysis in which the analytical properties of the trigonometrical functions are investigated.
www.1911encyclopedia.org /Trigonometry   (12282 words)

  
 Encyclopedia
ASTRONOMY, (qq.v.), in which the main problem generally was to determine an inaccessible distance, such as the distance between the earth and the moon, or of a distance that could not be measured directly, such as the distance across a large lake.
Trigonometry is also applied in physics, chemistry, and almost all branches of engineering, particularly in the study of periodic phenomena, such as vibration studies of sound, a bridge, or a building, or the flow of alternating current.
Spherical trigonometry, which is used principally in navigation and astronomy, is concerned with spherical triangles, that is, figures that are arcs of great circles (see
www.history.com /encyclopedia.do?vendorId=FWNE.fw..tr086900.a   (2687 words)

  
 Content
Trigonometry is the branch of mathematics which deals with the relations of sides and angles of triangles and with the relations among special functions associated with any angle.
Trigonometry was developed by ancient cultures as a tool to help with the precise mapping of the apparent motion of stars and planets through the sky, and with the prediction of celestial phenomena (faces of the moon, eclipses, equinoxes) with important effects on Earth.
As such, the spherical trigonometry (the study of spherical triangles on the surface of the "celestial sphere", and which have arcs of circumferences for sides), came first into existence.
web1.shastacollege.edu /cberisso/m2/m2_50101.htm   (414 words)

  
 Spherical trigonometry - Wikipedia, the free encyclopedia
To determine this, the spherical excess must be expressed in radians; the surface area A is then given in terms of the sphere's radius R by the expression:
The identity may be derived by considering the triangles formed by the tangent lines to the spherical triangle subtending angle C and using the plane law of cosines.
Spherical Trigonometry — for the use of colleges and schools by I. Todhunter, M.A., F.R.S. Historical Math Monograph posted by Cornell University Library.
en.wikipedia.org /wiki/Spherical_trigonometry   (742 words)

  
 Term paper on A History of the Development of Trigonometry
Trigonometry is associated with the study of the relationships that are found between the angles and the sides of the triangle.
Trigonometry has got the pride to be one of the most ancient subjects that were extremely famous all over the world and scholars from all over the world studied those ancient subjects.
The invention of trigonometry is associated with the geometric school of Alexandria that was well-known for the studies of astronomy.
www.termpapergenie.com /ahistory.html   (2529 words)

  
 Highbeam Encyclopedia - Search Results for trigonometry
trigonometry [Grmeasurement of triangles], a specialized area of geometry concerned with the properties of and relations among the parts of a triangle.
Spherical trigonometry is concerned with the study of triangles on the surface of a sphere rather than in the plane; it is of considerable importance in
He is one of the first known to have used algebra; his writings include rules of arithmetic and of plane and spherical trigonometry, and solutions of quadratic equations.
www.encyclopedia.com /SearchResults.aspx?Q=trigonometry   (394 words)

  
 Spherical Geometry and Trigonometry
Thus the area of a spherical triangle on a unit sphere is equal to the spherical exess, which is the sum of vertex angles in excess of π radians.
This relationship for the area of a spherical triangle generalizes to convex spherical polygons with the spherical excess being the sum of the angles - (n-2)π, where n is the number of sides of the polygon.
A side of a spherical triangle is the intersection of a plane passing through the center of a sphere with the surface of the sphere.
www.sjsu.edu /faculty/watkins/sphere.htm   (1395 words)

  
 Trigonometry?
Trigonometry is a branch of mathematics that developed from simple mensuration (measurement of geometric quantities), geometry, and surveying.
Trigonometry uses the fact that ratios of pairs of sides of triangles are functions of the angles.
The reduction formulas in trigonometry are identities that express trigonometric ratios of any angle in terms of ratios of acute angles.
omega.albany.edu:8008 /mat112dir/trig.html   (1095 words)

  
 Trigonometry
Since the proof of the spherical Law of Tangents is identical to that of the correspoinding plane Law of Tangents, we state the spherical Law of Tangents without proof.
An interesting problem of spherical trigonometry is that of finding the area of a spherical cap of either a cone or pyramid, with its apex at the center of the sphere.
For each law, we give the spherical (a, b, c are the sides; A, B, C are the angles), its dual for the polar triangle (A. C are the angles; a, b, c are the sides), and the plane (a, b, c are the norms of the sides; A, B, C are the angles) version.
www.rism.com /Trig/Trig02.htm   (8737 words)

  
 SPHERICAL TO PLANAR TRANSFORMATION   (Site not responding. Last check: 2007-11-06)
To illustrate the model, the icosehedral spherical triangle and the equilateral planar triangle are each divided into six similar triangles by projecting the three perpendicular bisectors.
To transform a point P on the Icosahedral spherical triangle, multiply the distance of the point P from the centroid, (=Dg from Rotation 3, Step 5), by the Amplitude factor and add the Angle factor to the bearing of the point P, (=B"' from Rotation 3, Step 7), from the perpendicular bisector.
This paper explained a scheme using spherical coordinate notation to transform the spherical facets of a regular icosahedron to planar facets suitable for plotting on an X-Y plotter.
www.academic.marist.edu /~jwg9/dymaxion/sphere.htm   (924 words)

  
 Applications of Trigonometry
The kind of trigonometry needed to understand positions on a sphere is called spherical trigonometry.
Spherical trigonometry is rarely taught now since its job has been taken over by linear algebra.
Of course, trigonometry is used throughout mathematics, and, since mathematics is applied throughout the natural and social sciences, trigonometry has many applications.
www.geocities.com /Hollywood/Academy/8245/trigonometry.html   (324 words)

  
 Spherical Trigonometry   (Site not responding. Last check: 2007-11-06)
One of the primary concerns in astronomy throughout history was the positioning of the heavenly bodies, for which spherical trigonometry was required.
The spherical triangle in the diagram above is the same as the trihedral, except that the sides a, b and c are great circle paths on the surface of the sphere centred at V rather than straight lines (as in the trihedral).
To solve a spherical triangle problem, known values are substituted into the equations, which are then solved for the remaining unknown values.
www.hps.cam.ac.uk /starry/sphertrig.html   (567 words)

  
 trigonometry - HighBeam Encyclopedia
Spherical trigonometry is concerned with the study of triangles on the surface of a sphere rather than in the plane; it is of considerable importance in surveying, navigation, and astronomy.
A general triangle, not necessarily containing a right angle, can also be analyzed by means of trigonometry, and various relationships are found to exist between the sides and angles of the general triangle.
This repeating, or periodic, nature of the trigonometric functions leads to important applications in the study of such periodic phenomena as light and electricity.
www.encyclopedia.com /doc/1E1-trigonom.html   (758 words)

  
 A Primer on Spherical Geometry   (Site not responding. Last check: 2007-11-06)
If trigonometry is a subject to send shudders of fear through the typical student (science majors included), then spherical geometry surely appears the devil's own invention.
The astronomical (or navigational) use for spherical trigonometry is to solve triangles on a spherical surface - either on the celestial sphere or on the surface of the Earth.
While spherical trig may seen either old fashioned or arcane, and it isn't taught much anymore, it is the basis of navigation.
www.ess.sunysb.edu /fwalter/AST443/sphgeo.html   (499 words)

  
 Sperry Spherical Trigonometry
There seems to be a real need for a short yet rigorous text in spherical trigonometry which shall contain little more theory than is needed for the solution of spherical triangles, with a few applications to add interest to the subject.
A knowledge of spherical geometry is not presupposed, but a brief presentation of the principal concepts and theorems is given in Chapter I. Most of the theoretical examples bear directly on the development of the theory in the text.
The time required for the discussion and solution of the general spherical triangle may be reduced by half by considering only Cases 1, 2, and 3 in Chapter III, explaining how the other cases may be solved by means of the polar triangle.
www.agnesscott.edu /lriddle/WOMEN/abstracts/sperry_spherical.htm   (453 words)

  
 Spherical Astronomy without Trig
Spherical astronomy concerns the directions of celestial objects, and uses the concept of the celestial sphere.
There is of course the heavy use of spherical trigonometry in most calculations.
The sperical analog of the straight line is the great circle, which if it were part of a plane, would cut the sphere in half.
ourworld.compuserve.com /homepages/bmoler/spheric.htm   (1288 words)

  
 Trigonometry and Basic Functions - Numericana
Spherical trigonometry: Triangles drawn on the surface of a sphere.
A spherical triangle is a figure on the surface of a sphere of radius R, featuring three sides which are arcs of great circles (a "great circle" is the intersection of the sphere with a plane containing the sphere's center).
The study of spherical triangles is often called spherical trigonometry and is about as ancient as the simpler planar trigonometry summarized above.
home.att.net /~numericana/answer/functions.htm   (4111 words)

  
 Trigonometry - Wikipedia, the free encyclopedia
Trigonometry specifically deals with the relationships between the sides and the angles of triangles, that is, the trigonometric functions, and with calculations based on these functions.
Trigonometry has important applications in many branches of pure mathematics as well as of applied mathematics and, consequently, much of science.
Using trigonometry and an accurate clock, the position of the ship can then be determined from several such measurements.
en.wikipedia.org /wiki/Trigonometry   (1525 words)

  
 Spherical Trigonometry
I used spherical trigonometry to calculate all the angles for the 48 and the 120 LCD spherical triangles of the vector equilibrium and the icosahedron generated by the primary great circles.
A spherical triangle is defined when three planes pass through the surface of a sphere and through the sphere's center of volume.
There are many methods for obtaining the solution to a spherical trigonometry problem as well as many other details which you will find in a spherical trigonometry text book.
www.rwgrayprojects.com /rbfnotes/trig/strig/strig.html   (745 words)

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