thumb|upright=1.5|Approximately equal behavior of some (trigonometric) functions for

For small angles, the trigonometric functions sine, cosine, and tangent can be calculated with reasonable accuracy by the following simple approximations:

<math display="block">

\begin{align}

\sin \theta &\approx \tan \theta \approx \theta, \\[5mu]

\cos \theta &\approx 1 - \tfrac12\theta^2 \approx 1,

\end{align}

</math>

provided the angle is measured in radians. Angles measured in degrees must first be converted to radians by multiplying them by .

These approximations have a wide range of uses in branches of physics and engineering, including mechanics, electromagnetism, optics, cartography, astronomy, and computer science.

Justifications

Geometric

frameless|upright=2

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

<math display="block"> \cos{\theta} \approx 1 - \frac{\theta^2}{2}.</math>

The opposite leg, , is approximately equal to the length of the blue arc, . The arc has length , and by definition and , and for a small angle, and , which leads to:

<math display="block">\sin \theta = \frac{O}{H}\approx\frac{O}{A} = \tan \theta = \frac{O}{A} \approx \frac{s}{A} = \frac{A\theta}{A} = \theta.</math>

Or, more concisely,

<math display="block">\sin \theta \approx \tan \theta \approx \theta.</math>

Calculus

Using the squeeze theorem,

<math display=block>\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}</math>

where is the angle in radians. For very small angles, higher powers of become extremely small, for instance if , then , just one ten-thousandth of . 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, , and drop the quadratic term and approximate the cosine as .

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

,

, and

.

Error of the approximations

thumb|300px|A graph of the [[relative errors for the small angle approximations (, , ) ]]

Near zero, the relative error of the approximations , , and is quadratic in : for each order of magnitude smaller the angle is, the relative error of these approximations shrinks by two orders of magnitude. The approximation has relative error which is quartic in : 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:

  • at about 0.14 radians (8.1°)
  • at about 0.17 radians (9.9°)
  • at about 0.24 radians (14.0°)
  • at about 0.66 radians (37.9°)

Slide-rule approximations

thumb | 300px | The left end of a [[Keuffel & Esser 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").]]

thumb | 300px | The right end of a K&E Decilon slide rule with a line to show the calibration of the SRT scale at 5.73 degrees.

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.

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%.

Angle sum and difference

The angle addition and subtraction theorems can be simplified when one of the angles is small (if is very small then and ):

<math display=block>\begin{align}

\cos(\alpha + \beta) &\approx \cos\alpha - \beta\sin\alpha, \\

\cos(\alpha - \beta) &\approx \cos\alpha + \beta\sin\alpha, \\

\sin(\alpha + \beta) &\approx \sin\alpha + \beta\cos\alpha, \\

\sin(\alpha - \beta) &\approx \sin\alpha - \beta\cos\alpha.

\end{align}</math>

Specific uses

Astronomy

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.

Optics

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

Wave interference

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 is the distance of a fringe from the center of maximum light intensity, is the order of the fringe, is the distance between the slits and projection screen, and is the distance between the slits: <math display="block">y \approx \frac{m\lambda D}{d}

</math>

Structural mechanics

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.

Piloting

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.

Interpolation

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

Example: sin(0.755)

<math display="block">\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}</math>

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.

See also

  • Skinny triangle
  • Versine
  • Exsecant

References

</references>