New Theorem and Formula for Circle Arc Length Calculations with Trigonometric Approach Application in Astrophysics

Abstract

The circle and sphere have been studied since the ancient Egyptians and Greeks before the Common Era (BCE). The recent scientific renaissance has also used them in different fields. It is also mentioned in the Prophet Mohamed`s Holy Quran. This article introduces a new Theorem (S. El-Mongy`s Theorem) as an empirical formula to correlate the constant (e) with circle and sphere. It states that “the arc length is correlated as a direct function in {(e π r s^A)}, whatever the central angle (ϴ) and radius (r). The factor s^Ais (ϴ/10ϕ). The formula can also be written as; A_L = {(0.0174533185 r ϴ)}. Where the value 0.0174533185 is a constant called Sayed`s number (I_s) and equals (eπ/10ϕ). This factor is very close to value (π/180 = 0.0174532925) with ~1.5x10^-4 % difference. The formula was applied for calculation the arc length (A_L) of circles of different radii and angles. The results of this formula were validated and verified for very wide range; from 0.5 cm to 4.4x10²³ km (46.5x10⁹ light-years; ly) and compared with the old published arc length formula results. The difference is from 0.000% to 0.002% only. The formula was also used as trigonometric functions of circular orbits for calculation the distances between the Earth and Sun, Moon, planets, stars, and EH-M87 Black hole with relatively small error; the difference is from 0.26% to maximum ~ 2.27%. The error was 0.29% for ~54 x 10⁶ ly distance to the M87 black hole. The S. El-Mongy formula may open the door for further scientific and engineering applications.