The basis of Trigonometry is the concept of right triangle congruence. The corresponding sides of two similarly built ups have the same proportions. On the geometry of Euclid, if each corner on two triangles has the same magnitude, then the two triangles must be identical.  This is the basis for comparison of trigonometric taper angles. This concept was then developed again for non-taper angles (over 90 degrees and less than zero degrees).
There are many trigonometric applications. It is primarily a triangulation technique used in astronomy to calculate the distance to nearby stars, in geography to calculate between specific points, and in satellite navigation systems.
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Other fields that use trigonometry include astronomy (and include navigation, at sea, air, and space), music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, biology, medical imaging (CAT scan and ultrasound, pharmacy, chemistry, number theory (and includes cryptology), seismology, meteorology, oceanography, various branches in physics, ground surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography.
There is a modern development of trigonometry that involves “spreading” and “quadrance”, not angle and length. This new approach is called rational trigonometry and is the work of Dr. Norman Wildberger of the University of New South Wales. More information can be found on its website.