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Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions.
Spherical geometry works similarly to Euclidean geometry in that there still exist points, lines, and angles. For instance, a "line" between two points on a sphere is actually a great circle of the sphere, which is also the projection of a line in three-dimensional space onto the sphere.
To derive the basic formulas pertaining to a spherical triangle, we use plane trigonometry on planes related to the spherical triangle. For example, planes tangent to the sphere at one of the vertices of the triangle, and central planes containing one side of the triangle.
The four formulas may be referred to as the sine formula, the cosine formula, the polar cosine formula, and the cotangent formula. Beneath each formula is shown a spherical triangle in which the four elements contained in the formula are highlighted.
In spherical geometry, angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects; for example, the sum of the interior angles of a spherical triangle exceeds 180 degrees.
1 Introduction. One of two standard non euclidean geometries. Hyperbolic geometry became fashionable because Thurston started it. Spherical geometry is still rather dormant. There are analogies between hyperbolic and spherical geometries.
Let a spherical triangle be drawn on the surface of a sphere of radius R, centered at a point O=(0,0,0), with vertices A, B, and C. The vectors from the center of the sphere to the vertices are therefore given by a=OA^->, b=OB^->, and c=OC^->.
To get started, let S be the sphere of radius 1 centered at the origin O in three dimensional space. Using xyz coordinates, we can place O at (0; 0; 0), which means x = y = z = 0, then S is given by x2 + y2 + z2 = 1. Points will now be understood as points on S. A line is now a great circle.
The study of figures on the surface of a sphere (such as the spherical triangle and spherical polygon), as opposed to the type of geometry studied in plane geometry or solid geometry. In spherical geometry, straight lines are great circles , so any two lines meet in two points.
Spherical trigonometry formulas. 1. Spherical Pythagorean theorem: c a b. cos = cos cos. R R R. 2. Formulas in a spherical right triangle. sin a tan. sin A = R. c sin and cos A = tan. R. 3. Spherical law of sines: sin a sin b c. R = R sin. = R sin(A) sin(B) sin(C) 4. Spherical law of cosines for sides: c a b a. cos = cos cos + sin sin. R R R R. 5.