Some brief descriptions of the Greek approach to trigonometry via chords of circles were given in the module Further trigonometry (Year 10). The chord equivalents of the double angle formulas and the sine and cosine expansions, \(\sin(A+B)\) and \(\cos(A+B)\), were found by Hipparchos (180--125 BCE) and Ptolemy (c. 90--168 CE).

The Indian mathematicians Aryabhata (476--550 CE) and Bhaskara I (7th century) reworked much of the Greek chord ideas and introduced ratios that are essentially the same as our sine and cosine. They were able to tabulate the values of these ratios. In particular, Bhaskara I used a formula equivalent to

\[ \sin x \approx \dfrac{16x (\pi - x)}{5 \pi^2 - 4x (\pi - x)}, \qquad \text{for } 0 \leq x \leq \dfrac{\pi}{2}, \] to produce tables of values.

In the 12th century, Bhaskara II discovered the sine and cosine expansions in roughly their modern form.

Arabic mathematicians were also working in this area and, in the 9th century, Muhammad ibn Musa al-Khwarizmi produced sine and cosine tables. He also gave a table of tangents.

The first mathematician in Europe to treat trigonometry as a distinct mathematical discipline was Regiomontanus. He wrote a treatise called De triangulis omnimodus in 1464, which is recognisably `modern' in its approach to the subject.

The history of the trigonometric functions really begins after the development of calculus and was largely developed by Leonard Euler (1707--1783). The works of James Gregory in the 17th century and Colin Maclaurin in the 18th century led to the development of infinite series expansions for the trigonometric functions, such as

\[ \sin \theta = \theta - \dfrac{\theta^3}{3!} + \dfrac{\theta^5}{5!} - \dotsb. \]

It was Euler who discovered the remarkable relationship between the exponential and trigonometric functions

\[ e^{i\theta} = \cos\theta + i\sin \theta, \] which is nowadays taken as a definition of \(e^{i\theta}\).

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