Contents
What is spherical excess of spherical triangle?
: the amount by which the sum of the three angles of a spherical triangle exceeds two right angles.
How do you calculate spherical excess?
On the plane, the sum of the interior angles of any triangle is exactly 180°. On a sphere, however, the corresponding sum is always greater than 180° but also less than 540°. That is, 180° < α + β + γ < 540° in the diagram above. The positive quantity E = α + β + γ – 180° is called the spherical excess of the triangle.
What is a spherical triangle called?
A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices. The spherical triangle is the spherical analog of the planar triangle, and is sometimes called an Euler triangle (Harris and Stocker 1998).
How do you solve a spherical triangle?
For right spherical triangles, it is customary to set C = 90°. One way of solving for the missing sides and angles of a right spherical triangle is using Napier’s rules. Napier’s rules consist of two parts, and are used in conjunction with a figure called Napier’s circle as shown.
Who is Father of spherical trigonometry?
Nasīr al-Dīn al-Tūsī
In the 13th century, Nasīr al-Dīn al-Tūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.
What are the properties of spherical triangle?
The three angles of a spherical triangle must together be more than 180° and less than 540° . 7. The greater side is opposite the greater angle , if tow sides are equal their opposite angles are equal . , and if one side of the triangle 90° it is called a quadrantal triangle .
What is the theorems on right spherical triangle?
SIN-TAAD Rule Spherical triangle can have one or two or three 90° interior angle. Spherical triangle is said to be right if only one of its included angle is equal to 90°. Triangles with more than one 90° angle are oblique.
Who invented trigonometry?
Hipparchus
Trigonometry in the modern sense began with the Greeks. Hipparchus (c. 190–120 bce) was the first to construct a table of values for a trigonometric function.
Why Sine is called sine?
The word “sine” (Latin “sinus”) comes from a Latin mistranslation by Robert of Chester of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.
Who is father of trigonometry?
mathematician Hipparchus
The first known table of chords was produced by the Greek mathematician Hipparchus in about 140 BC. Although these tables have not survived, it is claimed that twelve books of tables of chords were written by Hipparchus. This makes Hipparchus the founder of trigonometry.
Why do we teach trigonometry?
Great trigonometry skills allow students to work out complex angles and dimensions in relatively little time. Widely used in architecture, engineering and many sciences, trigonometry is one of the most valuable branches of mathematics.
What is the spherical excess of a planar triangle?
As we all know, the sum of angles of any planar triangle is equal to 180 degrees. However, for a spherical triangle, the sum of angles is always greater than 180 degrees. The difference between the sum of the spherical triangle angles and 180 degrees is called spherical excess .
Which is the best definition of spherical excess?
Definition of spherical excess : the amount by which the sum of the three angles of a spherical triangle exceeds two right angles
What is the area of a spherical triangle equal to?
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. The area of a spherical triangle on a sphere of radius r is equal to the spherical excess times r 2 .
Which is the best description of spherical trigonometry?
Spherical trigonometry. Spherical trigonometry is the branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons (especially spherical triangles) defined by a number of intersecting great circles on the sphere.