How is spherical trigonometry used in astronomy?
For astronomy, V is the position of the observer, and PQR are points on a sphere centred at V, the celestial sphere. These equations can be used to calculate positions on the celestial sphere, however, the laws can be simplified by consideration of the spherical triangle (see diagram below).
What is spherical triangle in astronomy?
1.3 Spherical triangle. A spherical triangle is the figure formed by arcs of great circle that pass by 3 points, connected by pairs, that intercept at the surface of a sphere (Fig. 1.9). An Eulerian spherical triangle has every and each side and angle less than 180 .
Who invented 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 is the difference between spherical geometry from spherical trigonometry?
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.
How many degrees are in a spherical triangle?
The sum of the angles of a triangle on a sphere is 180°(1 + 4f), where f is the fraction of the sphere’s surface that is enclosed by the triangle.
What is the Napier’s rule number 1 in solving the right spherical triangle?
Rule 1: The SINe of a missing part is equal to the product of the TAngents of its ADjacent parts (SIN-TA-AD rule).
What is the formula of spherical triangle?
Theorem: The area of spherical triangle △ABC is A + B + C − π. This quantity, A + B + C − π is called the excess. It’s amazing that the area has such a nice formula, depending only on the angles of the triangle!
What are the equations for spherical triangle?
Right spherical triangles
- sin a = sin α sin c = tan b cot β
- sin b = sin β sin c = tan a cot α
- cos α = cos a sin β = tan b cot c.
- cos β = cos b sin α = tan a cot c.
- cos c = cot α cot β = cos a cos b.
Why is spherical geometry important?
Spherical geometry is useful for accurate calculations of angle measure, area, and distance on Earth; the study of astronomy, cosmology, and navigation; and applications of stereographic projection throughout complex analysis, linear algebra, and arithmetic geometry.