For smallangles, thetrigonometric functions sine, cosine, and tangent can be calculated with reasonable accuracy by the following simple approximations:

provided the angle is measured inradians. Angles measured indegrees must first be converted to radians by multiplying them by⁠${\displaystyle \pi /180}$⁠.

These approximations have a wide range of uses in branches ofphysics andengineering, includingmechanics,electromagnetism,optics,cartography,astronomy, andcomputer science.^[1][2] One reason for this is that they can greatly simplifydifferential equations that do not need to be answered with absolute precision.

There are a number of ways to demonstrate the validity of the small-angle approximations. The most direct method is to truncate theMaclaurin series for each of the trigonometric functions. Depending on theorder of the approximation,${\displaystyle \cos \theta }$ is approximated as either${\displaystyle 1}$ or as$1-\frac{1}{2}{\theta }^2$.^[3]

For a small angle,H andA are almost the same length, and thereforecosθ is nearly 1. The segmentd (in red to the right) is the difference between the lengths of the hypotenuse,H, and the adjacent side,A, and has length${\displaystyle H-\sqrt{H^2-O^2}}$, which for small angles is approximately equal to${\displaystyle O^2/2H\approx \frac{1}{2}{\theta }^{2}H}$. As a second-order approximation,$${\displaystyle \cos \theta \approx 1-\frac{{\theta }^2}{2}.}$$

The opposite leg,O, is approximately equal to the length of the blue arc,s. The arcs has lengthθA, and by definitionsinθ =⁠O/H⁠ andtanθ =⁠O/A⁠, and for a small angle,O ≈s andH ≈A, which leads to:$${\displaystyle \sin \theta =\frac{O}{H}\approx \frac{O}{A}=\tan \theta =\frac{O}{A}\approx \frac{s}{A}=\frac{A\theta }{A}=\theta .}$$

Or, more concisely,$${\displaystyle \sin \theta \approx \tan \theta \approx \theta .}$$

Using thesqueeze theorem,^[4] we can prove that$${\displaystyle \underset{\theta \to 0}{lim}\frac{\sin (\theta )}{\theta }=1,}$$ which is a formal restatement of the approximation${\displaystyle \sin (\theta )\approx \theta }$ for small values ofθ.

A more careful application of the squeeze theorem proves that$${\displaystyle \underset{\theta \to 0}{lim}\frac{\tan (\theta )}{\theta }=1,}$$ from which we conclude that${\displaystyle \tan (\theta )\approx \theta }$ for small values ofθ.

Finally,L'Hôpital's rule tells us that$${\displaystyle \underset{\theta \to 0}{lim}\frac{\cos (\theta )-1}{{\theta }^2}=\underset{\theta \to 0}{lim}\frac{-\sin (\theta )}{2\theta }=-\frac{1}{2},}$$ which rearranges to$\cos (\theta )\approx 1-\frac{{\theta }^2}{2}$ for small values ofθ. Alternatively, we can use thedouble angle formula${\displaystyle \cos 2A\equiv 1-2{\sin }^{2}A}$. By letting${\displaystyle \theta =2A}$, we get that$\cos \theta =1-2{\sin }^2\frac{\theta }{2}\approx 1-\frac{{\theta }^2}{2}$.

TheTaylor series expansions of trigonometric functions sine, cosine, and tangent near zero are:^[5]

where⁠${\displaystyle \theta }$⁠ is the angle in radians. For very small angles, higher powers of⁠${\displaystyle \theta }$⁠ become extremely small, for instance if⁠${\displaystyle \theta =0.01}$⁠, then⁠${\displaystyle {\theta }^3=0.000001}$⁠, just one ten-thousandth of⁠${\displaystyle \theta }$⁠. 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,⁠${\displaystyle \sin \theta \approx \tan \theta \approx \theta }$⁠, and drop the quadratic term and approximate the cosine as⁠${\displaystyle \cos \theta \approx 1}$⁠.

If additional precision is needed the quadratic and cubic terms can also be included,⁠${\displaystyle \sin \theta \approx \theta -\frac{1}{6}{\theta }^3}$⁠,⁠${\displaystyle \cos \theta \approx 1-\frac{1}{2}{\theta }^2}$⁠, and⁠${\displaystyle \tan \theta \approx \theta +\frac{1}{3}{\theta }^3}$⁠.

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

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

• ⁠${\displaystyle \cos \theta \approx 1}$⁠ at about 0.14 radians (8.1°)
• ⁠${\displaystyle \tan \theta \approx \theta }$⁠ at about 0.17 radians (9.9°)
• ⁠${\displaystyle \sin \theta \approx \theta }$⁠ at about 0.24 radians (14.0°)
• ⁠${\displaystyle \cos \theta \approx 1-\frac{1}{2}{\theta }^2}$⁠ at about 0.66 radians (37.9°)

Manyslide 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.^[6]

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 0.1 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%.

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

cos(α +β)	≈ cos(α) −β sin(α),
cos(α −β)	≈ cos(α) +β sin(α),
sin(α +β)	≈ sin(α) +β cos(α),
sin(α −β)	≈ sin(α) −β cos(α).

Inastronomy, theangular size or angle subtended by the image of a distant object is often only a fewarcseconds (denoted by the symbol ″), so it is well suited to the small angle approximation.^[7] The linear size (D) is related to the angular size (X) and the distance from the observer (d) by the simple formula:

${\displaystyle D=X\frac{d}{206265{}^{″}}}$

whereX is measured in arcseconds.

The quantity206265″ is approximately equal to the number of arcseconds in acircle (1296000″), divided by2π, or, the number of arcseconds in 1 radian.

The exact formula is

${\displaystyle D=d\tan \left(X\frac{2\pi }{1296000{}^{″}}\right)}$

and the above approximation follows whentanX is replaced byX.

For example, theparsec is defined by the value of d whenD=1 AU,X=1 arcsecond, but the definition used is the small-angle approximation (the first equation above).

The second-order cosine approximation is especially useful in calculating thepotential energy of apendulum, which can then be applied with aLagrangian to find the indirect (energy) equation of motion. When calculating theperiod 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 describingsimple harmonic motion.^[8]

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

The sine and tangent small-angle approximations are used in relation to thedouble-slit experiment or adiffraction grating to develop simplified equations like the following, wherey is the distance of a fringe from the center of maximum light intensity,m is the order of the fringe,D is the distance between the slits and projection screen, andd is the distance between the slits:^[9]$${\displaystyle y\approx \frac{m\lambda D}{d}}$$

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

The1 in 60 rule used inair navigation has its basis in the small-angle approximation, plus the fact that one radian is approximately 60 degrees.

The formulas foraddition and subtraction involving a small angle may be used forinterpolating betweentrigonometric table values:

Example: sin(0.755)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.

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