Small-angle approximation#Motion of a pendulum

File:Small angle triangle.svg

For a small angle, {{mvar|H}} and {{mvar|A}} are almost the same length, and therefore {{math|cos θ}} is nearly 1. The segment {{mvar|d}} (in red to the right) is the difference between the lengths of the hypotenuse, {{mvar|H}}, and the adjacent side, {{mvar|A}}, and has length $\backslash textstyle\; H\; -\; \backslash sqrt\{H^2\; -\; O^2\}$, which for small angles is approximately equal to $\backslash textstyle\; O^2\backslash !/2H\; \backslash approx\; \backslash tfrac12\; \backslash theta^2H$. As a second-order approximation,

$$\backslash cos\{\backslash theta\}\; \backslash approx\; 1\; -\; \backslash frac\{\backslash theta^2\}\{2\}.$$

The opposite leg, {{mvar|O}}, is approximately equal to the length of the blue arc, {{mvar|s}}. The arc {{mvar|s}} has length {{math|1=θA}}, and by definition {{math|1=sin θ = {{sfrac|O|H}}}} and {{math|1=tan θ = {{sfrac|O|A}}}}, and for a small angle, {{math|O ≈ s}} and {{math|H ≈ A}}, which leads to:

$$\backslash sin\; \backslash theta\; =\; \backslash frac\{O\}\{H\}\backslash approx\backslash frac\{O\}\{A\}\; =\; \backslash tan\; \backslash theta\; =\; \backslash frac\{O\}\{A\}\; \backslash approx\; \backslash frac\{s\}\{A\}\; =\; \backslash frac\{A\backslash theta\}\{A\}\; =\; \backslash theta.$$

Or, more concisely,

$$\backslash sin\; \backslash theta\; \backslash approx\; \backslash tan\; \backslash theta\; \backslash approx\; \backslash theta.$$

Using the squeeze theorem, we can prove that

\lim_{\theta \to 0} \frac{\sin(\theta)}{\theta} = 1,

which is a formal restatement of the approximation $\backslash sin(\backslash theta)\; \backslash approx\; \backslash theta$ for small values of θ.

A more careful application of the squeeze theorem proves that $$$$

\lim_{\theta \to 0} \frac{\tan(\theta)}{\theta} = 1,

from which we conclude that $\backslash tan(\backslash theta)\; \backslash approx\; \backslash theta$ for small values of θ.

Finally, L'Hôpital's rule tells us that $$$$

\lim_{\theta \to 0} \frac{\cos(\theta)-1}{\theta^2} = \lim_{\theta \to 0} \frac{-\sin(\theta)}{2\theta} = -\frac{1}{2},

which rearranges to $\backslash cos(\backslash theta)\; \backslash approx\; 1\; -\; \backslash frac\{\backslash theta^2\}\{2\}$ for small values of θ. Alternatively, we can use the double angle formula $\backslash cos\; 2A\; \backslash equiv\; 1-2\backslash sin^2\; A$. By letting $\backslash theta\; =\; 2A$, we get that $\backslash cos\backslash theta=1-2\backslash sin^2\backslash frac\{\backslash theta\}\{2\}\backslash approx1-\backslash frac\{\backslash theta^2\}\{2\}$.

File:Small-angle approximation for sine function.svg

The Taylor series expansions of trigonometric functions sine, cosine, and tangent near zero are:{{cite book| authorlink=Mary L. Boas|last=Boas| first=Mary L.|title=Mathematical Methods in the Physical Sciences|year=2006| publisher=Wiley|page=26| isbn=978-0-471-19826-0}}

$$\backslash begin\{align\}$$

\sin \theta &= \theta - \frac16\theta^3 + \frac1{120}\theta^5 - \cdots, \\[6mu]

\cos \theta &= 1 - \frac1{2}{\theta^2} + \frac1{24}\theta^4 - \cdots, \\[6mu]

\tan \theta &= \theta + \frac{1}{3}\theta^3 + \frac{2}{15}\theta^5 + \cdots.

\end{align}

where {{tmath|\theta}} is the angle in radians. For very small angles, higher powers of {{tmath|\theta}} become extremely small, for instance if {{tmath|1= \theta = 0.01}}, then {{tmath|1= \theta^3 = 0.000\,001}}, just one ten-thousandth of {{tmath|\theta}}. Thus for many purposes it suffices to drop the cubic and higher terms and approximate the sine and tangent of a small angle using the radian measure of the angle, {{tmath|\sin\theta \approx \tan\theta \approx \theta}}, and drop the quadratic term and approximate the cosine as {{tmath|\cos\theta \approx 1}}.

If additional precision is needed the quadratic and cubic terms can also be included,

{{tmath|\sin\theta \approx \theta - \tfrac16\theta^3}},

{{tmath|\cos\theta \approx 1 - \tfrac12\theta^2}}, and

{{tmath|\tan\theta \approx \theta + \tfrac13\theta^3}}.

One may also use dual numbers, defined as numbers in the form $a\; +\; b\backslash varepsilon$, with $a,b\backslash in\backslash mathbb\; R$ and $\backslash varepsilon$ satisfying by definition $\backslash varepsilon^2\; =\; 0$ and $\backslash varepsilon\; \backslash ne\; 0$. By using the MacLaurin series of cosine and sine, one can show that $\backslash cos(\backslash theta\backslash varepsilon)\; =\; 1$ and $\backslash sin(\backslash theta\backslash varepsilon)\; =\; \backslash theta\backslash varepsilon$. Furthermore, it is not hard to prove that the Pythagorean identity holds:$$\backslash sin^2(\backslash theta\backslash varepsilon)\; +\; \backslash cos^2(\backslash theta\backslash varepsilon)\; =\; (\backslash theta\backslash varepsilon)^2\; +\; 1^2\; =\; \backslash theta^2\backslash varepsilon^2\; +\; 1\; =\; \backslash theta^2\; \backslash cdot\; 0\; +\; 1\; =\; 1$$

File:Small angle compare error.svgs for the small angle approximations ({{tmath|\tan \theta \approx \theta}}, {{tmath|\sin \theta \approx \theta}}, {{tmath|\textstyle \cos \theta \approx 1 - \tfrac12\theta^2 }}) ]]

Near zero, the relative error of the approximations {{tmath|\cos \theta \approx 1}}, {{tmath|\sin \theta \approx \theta}}, and {{tmath|\tan \theta \approx \theta}} is quadratic in {{tmath|\theta}}: for each order of magnitude smaller the angle is, the relative error of these approximations shrinks by two orders of magnitude. The approximation {{tmath|\textstyle \cos \theta \approx 1 - \tfrac12\theta^2 }} has relative error which is quartic in {{tmath|\theta}}: for each order of magnitude smaller the angle is, the relative error shrinks by four orders of magnitude.

Figure 3 shows the relative errors of the small angle approximations. The angles at which the relative error exceeds 1% are as follows:

• {{tmath|\cos \theta \approx 1 }} at about 0.14 radians (8.1°)
• {{tmath|\tan \theta \approx \theta}} at about 0.17 radians (9.9°)
• {{tmath|\sin \theta \approx \theta}} at about 0.24 radians (14.0°)
• {{tmath|\textstyle \cos \theta \approx 1 - \tfrac12\theta^2 }} at about 0.66 radians (37.9°)

File:K&E Decilon slide rule left end with SRT scale.jpg Deci-Lon slide rule, with a thin blue line added to show the values on the S, T, and SRT scales corresponding to sine and tangent values of 0.1 and 0.01. The S scale shows arcsine(0.1) = 5.74 degrees; the T scale shows arctangent(0.1) = 5.71 degrees; the SRT scale shows arcsine(0.01) = arctangent(0.01) = 0.01*180/pi = 0.573 degrees (to within "slide-rule accuracy").]]

File:K&E Decilon slide rule right end with line.jpg

Many slide rules – especially "trig" and higher models – include an "ST" (sines and tangents) or "SRT" (sines, radians, and tangents) scale on the front or back of the slide, for computing with sines and tangents of angles smaller than about 0.1 radian.{{cite book |title=Communications Technician M 3 & 2 |date=1965 |publisher=Bureau of Naval Personnel |page=481 |url=https://books.google.com/books?id=FYB3o7iGvb8C&pg=PA481#v=onepage&q&f=false |access-date=7 March 2025}}

The right-hand end of the ST or SRT scale cannot be accurate to three decimal places for both arcsine(0.1) = 5.74 degrees and arctangent(0.1) = 5.71 degrees, so sines and tangents of angles near 5 degrees are given with somewhat worse than the usual expected "slide-rule accuracy". Some slide rules, such as the K&E Deci-Lon in the photo, calibrate to be accurate for radian conversion, at 5.73 degrees (off by nearly 0.4% for the tangent and 0.2% for the sine for angles around 5 degrees). Others are calibrated to 5.725 degrees, to balance the sine and tangent errors at below 0.3%.

The angle addition and subtraction theorems reduce to the following when one of the angles is small (β ≈ 0):

:

In astronomy, the angular size or angle subtended by the image of a distant object is often only a few arcseconds (denoted by the symbol ″), so it is well suited to the small angle approximation. The linear size ({{mvar|D}}) is related to the angular size ({{mvar|X}}) and the distance from the observer ({{mvar|d}}) by the simple formula:

:$D\; =\; X\; \backslash frac\{d\}\{206\backslash ,265\{\text{'}\text{'}\}\}$

where {{mvar|X}} is measured in arcseconds.

The quantity {{val|206265|u="}} is approximately equal to the number of arcseconds in a circle ({{val|1296000|u="}}), divided by {{math|2π}}, or, the number of arcseconds in 1 radian.

The exact formula is

:$D\; =\; d\; \backslash tan\; \backslash left(\; X\; \backslash frac\{2\backslash pi\}\{1\backslash ,296\backslash ,000\{\text{'}\text{'}\}\}\; \backslash right)$

and the above approximation follows when {{math|tan X}} is replaced by {{mvar|X}}.

For example, the parsec is defined by the value of d when {{mvar|D}}=1 AU, {{mvar|X}}=1 arcsecond, but the definition used is the small-angle approximation (the first equation above).

{{main|Pendulum (mechanics)#Small-angle approximation}}

The second-order cosine approximation is especially useful in calculating the potential energy of a pendulum, which can then be applied with a Lagrangian to find the indirect (energy) equation of motion. When calculating the period of a simple pendulum, the small-angle approximation for sine is used to allow the resulting differential equation to be solved easily by comparison with the differential equation describing simple harmonic motion.{{cite book |last1=Baker |first1=Gregory L. |last2=Blackburn |first2=James A. |chapter=Pendulums somewhat simple |title=The Pendulum: A Case Study in Physics |publisher=Oxford |year=2005 |doi=10.1093/oso/9780198567547.003.0002 |at=Ch. 2, {{pgs|8–26}} |isbn=0-19-856754-5 |chapter-url=https://archive.org/details/pendulumcasestud0000bake/page/8/mode/2up |chapter-url-access=limited }} {{pb}} {{cite journal |last=Bissell |first=John J. |year=2025 |title=Proof of the small angle approximation {{tmath|\sin \theta \approx \theta}} using the geometry and motion of a simple pendulum |journal=International Journal of Mathematical Education in Science and Technology |volume=56 |number=3 |pages=548–554 |doi=10.1080/0020739X.2023.2258885 |doi-access=free }}

In optics, the small-angle approximations form the basis of the paraxial approximation.

The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to develop simplified equations like the following, where {{mvar|y}} is the distance of a fringe from the center of maximum light intensity, {{mvar|m}} is the order of the fringe, {{mvar|D}} is the distance between the slits and projection screen, and {{mvar|d}} is the distance between the slits: {{Cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/slits.html|title=Slit Interference}}$$y\; \backslash approx\; \backslash frac\{m\backslash lambda\; D\}\{d\}$$

The small-angle approximation also appears in structural mechanics, especially in stability and bifurcation analyses (mainly of axially-loaded columns ready to undergo buckling). This leads to significant simplifications, though at a cost in accuracy and insight into the true behavior.

The 1 in 60 rule used in air navigation has its basis in the small-angle approximation, plus the fact that one radian is approximately 60 degrees.

The formulas for addition and subtraction involving a small angle may be used for interpolating between trigonometric table values:

Example: sin(0.755)

$$\backslash begin\{align\}$$

\sin(0.755) &= \sin(0.75 + 0.005) \\

& \approx \sin(0.75) + (0.005) \cos(0.75) \\

& \approx (0.6816) + (0.005)(0.7317) \\

& \approx 0.6853.

\end{align}

where the values for sin(0.75) and cos(0.75) are obtained from trigonometric table. The result is accurate to the four digits given.

• Skinny triangle
• Versine
• Exsecant

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{{DEFAULTSORT:Small-Angle Approximation}}

Category:Trigonometry

Category:Equations of astronomy