# How were sine and cosine functions computed before the notion of Taylor series?

We know using modern analysis techniques that $$\sin x$$ and $$\cos x$$ can be computed by their Taylor series (in fact the Taylor series are given as the definitions of these functions in today's real analysis books).

What I am wondering is, how were $$\cos x$$ and $$\sin x$$ functions computed before the notion of Taylor series was invented/discovered? Was it simple done by measurement and logging results into tables? I'd be very interested to know.

I am aware that people worked with functions that are related to but not the same as $$\sin x$$ and $$\cos x$$. The answers in this post explain that the chord function was used before sine and cosine, but my question still stands. All the trigonometric functions are related to one another, and one can equally ask, how was e.g. the chord function computed before the notion of Taylor series?

• By using half-angle, angle-sum and angle-difference relations (or their geometric equivalents), inequalities and linear interpolation, see How Ptolemy computed chords and Aaboe, Episodes from the Early History of Mathematics, ch. 4 for details. Oct 8, 2022 at 6:25
• The earliest known trigonometric tables were published in a multi-volume work by Hipparchus (c. 190-120 BC) around 140 BC. He tabulated chords, probably using Pythagorean and half-angle formulas. But as the work is no longer extant, we do not know for sure. Aryabhata (476–550 AD) was the first to publish a table of sines around 500 AD, probably using the Pythagorean formula and the half-angle formula $\sin \frac{A}{2} = \frac{1}{2}\sqrt{\sin^2 A + \mathrm{vers}^2 A}$, where $\mathrm{vers}$ is the versine Oct 8, 2022 at 6:37
• Ptolemy (c. 100- 170 AD) supposedly was aware of the angle-sum formula (here stated in modern terms): $\sin(x + y) = \sin x \cos y + \cos x \sin y$, while Abū al-Wafāʾ (940-998 AD), who compiled tables of sines and tangents at 15' intervals, used $\sin 2x = 2 \sin x \cos x$. Indian mathematician devised interpolation formulas up to third order during the 11ᵗʰ-12ᵗʰ centuries, and by the 15ᵗʰ-16ᵗʰ centuries had developed what are basically equivalents to the Taylor expansion and Maclaurin series. Oct 8, 2022 at 6:54
• Power series expansion is actually a poor method, and even at this time trigonometric functions are computed by a different method. Oct 10, 2022 at 14:46
• From the Wikipedia article on Rheticus "Valentinus Otho, devoted to completing his teacher's work, oversaw the hand computation of approximately 100,000 ratios to at least ten decimal places." Clearly, this was not the result of measurements. Oct 14, 2022 at 0:55