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Every time your physics teacher writes sin θ ≈ θ for a pendulum, they are using the first term of a series published in 1715 by an English mathematician called Brook Taylor. Most students learn the shortcut without being told where it comes from or when it stops working. This post covers both: who Taylor was, what his series says, and how to check an approximation with real numbers.
Who was Brook Taylor?
Taylor was born on 18 August 1685 in Edmonton, near London, and died on 29 December 1731, according to his MacTutor biography. He was taught at home by private tutors, entered St John's College, Cambridge, in 1703, and was elected to the Royal Society on 3 April 1712. From 1714 to 1718 he served as the Society's Secretary.
MacTutor credits him with three lasting ideas: the calculus of finite differences, integration by parts, and the series that carries his name. All three appear in his 1715 book *Methodus incrementorum directa et inversa*. The same year he published *Linear Perspective*, a mathematical treatment of vanishing points for painters, which drew on the painting and music he had loved since childhood.
Did Taylor invent the Taylor series?
Not alone, and he did not claim to. MacTutor lists James Gregory, Newton, Leibniz, Johann Bernoulli and Abraham de Moivre as mathematicians who had found versions of the result independently. Taylor first described his version in a letter to John Machin dated 26 July 1712, and explained that the idea came from a conversation in Child's Coffeehouse about using "Sir Isaac Newton's series" to solve Kepler's problem.
The result was not widely valued at first. MacTutor notes that it was Lagrange, in 1772, who called it the basic principle of the differential calculus, and the name "Taylor's series" came into use decades after Taylor's death. A detailed study of the book, Brook Taylor and the method of increments (Feigenbaum, 1985), runs to 140 pages in *Archive for History of Exact Sciences*.
What does the Taylor series say?
If you know a function and all its derivatives at one point a, you can estimate it at a nearby point a + h:
f(a + h) = f(a) + h f′(a) + (h²/2!) f″(a) + (h³/3!) f‴(a) + ...
Each term is smaller than the last when h is small, so a few terms usually give an excellent estimate. When a = 0 the series is often called a Maclaurin series.
Apply it to sin x at a = 0. The derivatives cycle: sin 0 = 0, cos 0 = 1, −sin 0 = 0, −cos 0 = −1. So:
sin x ≈ x − x³/6 + x⁵/120 − ...
The first term alone is the small-angle approximation. Your NCERT Class 11 Physics chapter on oscillations uses it to show that a simple pendulum moves in simple harmonic motion, and its Table 13.1 shows that sin θ stays close to θ (in radians) for angles as large as 20 degrees.
Worked example: how good is sin θ ≈ θ at 20°?
Convert to radians. 20° × π/180 = 0.3491 rad.
True value. sin 20° = 0.3420.
First term only. sin θ ≈ 0.3491. This is too big by 0.0071, an error of about 2.1%.
Add the cubic term. θ³/6 = (0.3491)³/6 = 0.04253/6 = 0.00709. So sin θ ≈ 0.3491 − 0.0071 = 0.3420, an error of about 0.01%.
At 10° (0.1745 rad) the first term alone is off by only about 0.5%. That is why textbooks say "small" angles: the error grows roughly with the square of the angle.
The same idea gives another shortcut you meet often. For small x, √(1 + x) ≈ 1 + x/2. So √1.02 ≈ 1.01, and the true value is 1.00995. Our guide to the binomial theorem for JEE covers the related expansion for whole-number powers.
Common mistakes with series approximations
Using degrees. The series for sin x only works with x in radians. NCERT's Oscillations chapter makes the same point: sin(15) means the sine of 15 radians, not 15 degrees.
Forgetting the factorials. The cubic term is x³/6 because 3! = 6. Writing x³/3 doubles the correction and makes the answer worse than the first term alone.
Approximating when x is not small. At 60° (1.047 rad), sin θ ≈ θ is wrong by about 21%. Check the size of x first.
Treating the syllabus as the limit. The JEE (Main) 2026 syllabus does not name Taylor series, so you will not be asked to derive one, but physics questions still expect you to use the approximations it produces. Check the syllabus for your exam year.
If a step like "for small θ" in a solution leaves you unsure why it is allowed, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can read the idea behind the approximation before you rely on it.
What can a student learn from how Taylor worked?
Taylor took a practical problem from a coffeehouse conversation, solving Kepler's equation, and turned it into a general method. He also checked his ideas in several fields, from vibrating strings to perspective drawing. The habit worth copying is to ask, every time you use a shortcut, what general rule it comes from and what error it hides.
Practice routine for this week:
Compute sin x, cos x and eˣ at x = 0.1, 0.3 and 0.5 using two terms of each series, and compare with a calculator.
For each, write the size of the first dropped term next to your error. Notice how closely they match.
Find three places in your physics notes where "small angle" or "x ≪ 1" appears, and write which series term was dropped.
Tonight, take one pendulum problem and check how large the angle could be before the small-angle answer is off by 1%.
