It can be solved by the important limit 1: lim (x→ 0) tan5x/x = 5l im (x→ 0) tan5x/(5x) = 5. Limit is a transformation based on the basic formula of trigonometric function, and the common ones are: (1) equivalent infinitesimal substitution, (2) Lobida rule.

Development history

origin

From the 5th century to12nd century, Indian mathematicians made great contributions to trigonometry. Although trigonometry was still a computing tool and an accessory of astronomy at that time, the content of trigonometry was greatly enriched through the efforts of Indian mathematicians.

The concepts of sine and cosine in trigonometry were first introduced by Indian mathematicians, who also made sine tables more accurate than Ptolemy.

As we already know, the chord table created by Ptolemy and Hipparchus is a circular full chord table, which corresponds to arcs and chords sandwiched between arcs. Unlike Indian mathematicians, they correspond the half chord (AC) to the half arc (AD) of the whole chord, that is, AC corresponds to ∠AOC. In this way, they created a sine table instead of a full chord table.