Hubbry Logo
Small-angle approximationSmall-angle approximationMain
Open search
Small-angle approximation
Community hub
Small-angle approximation
logo
8 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Small-angle approximation
Small-angle approximation
Small-angle approximation
View on Wikipedia
from Wikipedia
Approximately equal behavior of some (trigonometric) functions for x → 0

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

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.[1][2] One reason for this is that they can greatly simplify differential 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 the Maclaurin series for each of the trigonometric functions. Depending on the order of the approximation, is approximated as either or as .[3]

Justifications

[edit]

Geometric

[edit]

For a small angle, H and A are almost the same length, and therefore cos θ is nearly 1. The segment d (in red to the right) is the difference between the lengths of the hypotenuse, H, and the adjacent side, A, and has length , which for small angles is approximately equal to . As a second-order approximation,

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

Or, more concisely,

Calculus

[edit]

Using the squeeze theorem,[4] we can prove that which is a formal restatement of the approximation for small values of θ.

A more careful application of the squeeze theorem proves that from which we conclude that for small values of θ.

Finally, L'Hôpital's rule tells us that which rearranges to for small values of θ. Alternatively, we can use the double angle formula . By letting , we get that .

Algebraic

[edit]
The small-angle approximation for the sine function.

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

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

[edit]
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

[edit]
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").
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.[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%.

Angle sum and difference

[edit]

The angle 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).

Specific uses

[edit]

Astronomy

[edit]

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.[7] The linear size (D) is related to the angular size (X) and the distance from the observer (d) by the simple formula:

where X is measured in arcseconds.

The quantity 206265″ is approximately equal to the number of arcseconds in 1 radian, which is the number of arcseconds in a circle (1296000″) divided by 2π.

The exact formula is

and the above approximation follows when tan X is replaced by X.

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

Motion of a pendulum

[edit]
Main article: 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.[8]

Optics

[edit]

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

Wave interference

[edit]

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 y 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, and d is the distance between the slits: [9]

Structural mechanics

[edit]

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

[edit]

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

[edit]

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

See also

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Small-angle approximation

View on Grokipedia
from Grokipedia
The small-angle approximation is a fundamental mathematical technique used in physics and to simplify calculations involving when the angle θ is small and measured in radians. For such angles, typically θ ≪ 1 (e.g., less than about 0.1 radians or 6°), the approximations sin θ ≈ θ, tan θ ≈ θ, and cos θ ≈ 1 hold with high accuracy, as derived from the first few terms of the expansions of these functions around θ = 0. These relations stem from the power series representations: sin θ = θ - θ³/6 + ..., cos θ = 1 - θ²/2 + ..., and tan θ = θ + θ³/3 + ..., where higher-order terms become negligible for small θ. The approximation introduces errors on the order of θ³ or smaller, often less than 0.2% for θ < 0.1 radians, making it practical for many analytical solutions. This approximation originates from the limiting behavior of trigonometric functions as θ approaches zero, where the derivative of sin θ at 0 is cos 0 = 1, yielding the tangent line y = θ as a linear approximation. It is particularly valuable in contexts where exact solutions are intractable, such as deriving the simple harmonic motion equation for , where the restoring torque is linearized to sin θ ≈ θ, leading to a period independent of amplitude for small oscillations. In optics, it underpins the for ray tracing in lenses and mirrors, assuming rays are close to the optical axis with small angles to enable linear equations. Beyond these, the small-angle approximation facilitates computations in astronomy for estimating object sizes from angular diameters, using θ ≈ D/d where D is the physical diameter and d the distance, especially when d ≫ D. It also appears in wave mechanics, vector analysis, and numerical simulations, where it reduces nonlinear problems to linear ones for faster convergence and insight. While powerful, its validity diminishes for larger angles, necessitating full trigonometric evaluations or higher-order expansions in precise modeling.

Fundamentals

Definition and Basic Approximations

The small-angle approximation refers to the simplification of trigonometric functions and related expressions when the angle θ is much smaller than 1 radian, typically satisfying θ ≪ 1, allowing higher-order terms in their series expansions to be neglected for practical computations. This approach is particularly useful in and where exact trigonometric evaluations are cumbersome, providing a linear or quadratic estimate that maintains sufficient accuracy for small deviations from zero. The primary approximations for the basic trigonometric functions are derived from their Taylor series expansions around θ = 0. Specifically, for θ in radians, sin θ ≈ θ, cos θ ≈ 1 - (θ²/2), and tan θ ≈ θ. These relations hold because the leading terms dominate when θ is small, with the sine and tangent functions approximating the angle itself linearly, while cosine deviates quadratically from unity. From these, derived approximations for the reciprocal functions follow directly: sec θ ≈ 1 + (θ²/2), csc θ ≈ 1/θ, and cot θ ≈ 1/θ - (θ/3). The use of radians is essential for these approximations, as the Taylor series coefficients are dimensionless only in this unit system, ensuring the approximations scale correctly without additional conversion factors. These approximations trace their origins to early 18th-century developments in calculus, notably the infinite series expansions introduced by in his 1715 work Methodus Incrementorum Directa et Inversa and specialized by in his 1742 treatise Treatise of Fluxions, which formalized the basis for truncating series at low orders for small arguments.

Assumptions and Units

The small-angle approximation relies on the fundamental assumption that the angle θ is sufficiently small relative to the scale of the problem, typically in the limit as θ approaches zero, where higher-order terms in the series expansion become negligible. This condition ensures that perturbations or deviations from the idealized small-angle regime do not significantly affect the accuracy, as the approximation is asymptotic in nature—its precision improves monotonically as θ decreases toward zero. A critical requirement for the numerical validity of these approximations is that angles must be expressed in radians, the natural unit for trigonometric functions derived from their Taylor series expansions around θ = 0. Using degrees without conversion introduces substantial errors because the series coefficients are calibrated specifically for the radian measure, where the arc length equals the radius for θ = 1; in degrees, the equivalent "small" angle would require rescaling by π/180, rendering the direct substitution sin θ ≈ θ invalid. For reference, 1 radian is approximately 57.3 degrees, highlighting why unadjusted degree-based approximations fail to capture the linear behavior near zero. The approximation sin θ ≈ θ has a relative error of approximately 1% at θ ≈ 0.25 radians (about 14°), with error increasing for larger angles. For example, at θ = 0.1 radians (≈ 5.7°), the percentage error is approximately 0.17%, calculated as |(sin 0.1 - 0.1)/sin 0.1| × 100. At θ = 0.22 radians (≈ 12.6°), the error rises to about 0.81%, with sin(0.22) ≈ 0.2182 versus the approximation 0.22. These examples illustrate the rapid improvement in accuracy for smaller angles, underscoring the approximation's utility in contexts demanding high precision for θ ≪ 1 radian.

Justifications

Geometric Justification

The small-angle approximation for sine arises from considering a unit circle, where an angle θ\theta (in radians) subtends an arc of length θ\theta. In this setup, the chord length opposite the angle is sin⁡θ\sin \theta, and as θ\theta approaches zero, the arc length and chord length become indistinguishable, leading to sin⁡θ≈θ\sin \theta \approx \theta. This can be visualized by inscribing a sector with a small angle θ\theta at the center; the vertical rise along the ray equals the arc length in the limit, confirming the approximation geometrically without relying on series expansions. A more rigorous geometric justification uses the squeeze theorem on areas within the unit circle diagram. Consider three regions sharing the origin: a small triangle with area 12cos⁡θsin⁡θ\frac{1}{2} \cos \theta \sin \theta, the circular sector (wedge) with area 12θ\frac{1}{2} \theta, and a larger triangle with area 12tan⁡θ\frac{1}{2} \tan \theta. Since the small triangle is contained within the sector, which is contained within the large triangle, the area inequalities yield cos⁡θ≤sin⁡θθ≤1cos⁡θ\cos \theta \leq \frac{\sin \theta}{\theta} \leq \frac{1}{\cos \theta} for 0<θ<π20 < \theta < \frac{\pi}{2}. As θ→0\theta \to 0, both bounds approach 1, squeezing lim⁡θ→0sin⁡θθ=1\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1, thus sin⁡θ≈θ\sin \theta \approx \theta for small θ\theta. For cosine, the approximation cos⁡θ≈1\cos \theta \approx 1 follows from the same unit circle geometry. The point on the circle at angle θ\theta has x-coordinate cos⁡θ\cos \theta, which represents the horizontal projection from the origin. As θ\theta shrinks toward zero, this projection approaches the full radius of 1, with the vertical offset sin⁡θ\sin \theta becoming negligible; the "sagitta" (curved deviation from the straight line) vanishes proportionally to θ2\theta^2, leaving cos⁡θ\cos \theta arbitrarily close to 1. The tangent approximation tan⁡θ≈θ\tan \theta \approx \theta emerges from a right triangle inscribed in the unit circle, where the opposite side over the adjacent side (of length 1) gives tan⁡θ\tan \theta. For small θ\theta, the ray at angle θ\theta intersects the vertical line at x=1 (the tangent line to the circle at (1,0)) at a height approximately equal to the arc length θ\theta, since the circle and the tangent line coincide near the x-axis; thus, the opposite side approaches θ\theta. This is evident in the skinny triangle formed by the origin, the point (1,0), and the intersection point (1, \tan \theta), where the hypotenuse closely hugs the adjacent side as θ→0\theta \to 0. In skinny triangles—narrow right triangles with a small apex angle θ\theta and long equal sides—the hypotenuse approximates the adjacent side, reinforcing sin⁡θ≈tan⁡θ≈θ\sin \theta \approx \tan \theta \approx \theta. As θ\theta decreases, the differences between the arc length, chord (sine), and tangent segment diminish proportionally, illustrating the unified geometric limit where all three measures converge for infinitesimal angles.

Calculus-Based Justification

The small-angle approximation for trigonometric functions arises from the Taylor series expansions of these functions around θ = 0, where higher-order terms become negligible for small values of θ (typically in radians). The Taylor series, also known as the Maclaurin series when expanded about zero, represents a function as an infinite sum of terms calculated from its derivatives at that point. For the sine function, the Maclaurin series is sin⁡θ=θ−θ33!+θ55!−θ77!+⋯ ,\sin \theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \frac{\theta^7}{7!} + \cdots, where the series alternates signs and involves odd powers of θ. For small θ, terms involving θ³ and higher powers are much smaller than θ, so truncating after the first term yields sin⁡θ≈θ\sin \theta \approx \theta. Similarly, the Maclaurin series for cosine is cos⁡θ=1−θ22!+θ44!−θ66!+⋯ ,\cos \theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \frac{\theta^6}{6!} + \cdots, featuring even powers of θ with alternating signs. Neglecting terms beyond the constant and quadratic yields cos⁡θ≈1−θ22\cos \theta \approx 1 - \frac{\theta^2}{2}, which is the leading approximation for small angles. This expansion justifies the small-angle behavior where cosine is close to unity, with a small quadratic correction. The tangent function's approximation follows from the ratio of sine to cosine. Substituting the series gives tan⁡θ=sin⁡θcos⁡θ≈θ1−θ22\tan \theta = \frac{\sin \theta}{\cos \theta} \approx \frac{\theta}{1 - \frac{\theta^2}{2}}. For small θ, the denominator can be expanded using the binomial approximation (1−u)−1≈1+u(1 - u)^{-1} \approx 1 + u where u=θ22u = \frac{\theta^2}{2}, yielding θ1−θ22≈θ(1+θ22)≈θ+θ32\frac{\theta}{1 - \frac{\theta^2}{2}} \approx \theta \left(1 + \frac{\theta^2}{2}\right) \approx \theta + \frac{\theta^3}{2}. Truncating at the linear term provides tan⁡θ≈θ\tan \theta \approx \theta, consistent with the first-order approximations for sine and cosine. The full Maclaurin series for tangent is tan⁡θ=θ+13θ3+215θ5+17315θ7+⋯\tan \theta = \theta + \frac{1}{3}\theta^3 + \frac{2}{15}\theta^5 + \frac{17}{315}\theta^7 + \cdots. The validity of these truncations is supported by Taylor's theorem, which includes a remainder term estimating the error after n terms. In Lagrange's form, the remainder Rn(θ)R_n(\theta) after the polynomial of degree n is Rn(θ)=f(n+1)(ξ)(n+1)!θn+1R_n(\theta) = \frac{f^{(n+1)}(\xi)}{(n+1)!} \theta^{n+1}, where ξ is between 0 and θ. For sine approximated by the degree 1 polynomial θ (n=1), R1(θ)=f′′(ξ)2!θ2=−sin⁡ξ2θ2R_1(\theta) = \frac{f''(\xi)}{2!} \theta^2 = -\frac{\sin \xi}{2} \theta^2, bounded by θ22\frac{\theta^2}{2} in absolute value since |\sin \xi| \leq 1, which diminishes as θ approaches zero. Similar bounds apply to cosine and tangent, confirming that the approximations improve as θ becomes small, with the error scaling with higher powers of θ. These series expansions were formalized by Brook Taylor in his 1715 work Methodus Incrementorum Directa et Inversa, introducing the general theorem, and further developed by Colin Maclaurin in his 1742 Treatise on Fluxions, which applied them systematically to functions like sine and cosine.

Algebraic Justification

The small-angle approximation for the sine function can be justified algebraically through the limit lim⁡θ→0sin⁡θθ=1\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1, which holds when θ\theta is measured in radians and implies sin⁡θ≈θ\sin \theta \approx \theta for small θ\theta. This limit is established using the squeeze theorem applied to geometric inequalities derived from the unit circle, combined with basic trigonometric identities such as sin⁡θ<θ<tan⁡θ\sin \theta < \theta < \tan \theta for 0<θ<π/20 < \theta < \pi/2. An intuitive algebraic reinforcement of this approximation involves iterative application of the double-angle identity sin⁡θ=2sin⁡(θ/2)cos⁡(θ/2)\sin \theta = 2 \sin(\theta/2) \cos(\theta/2). For sufficiently small θ\theta, the half-angle θ/2\theta/2 is even smaller, where cos⁡(θ/2)≈1\cos(\theta/2) \approx 1 holds approximately, yielding sin⁡θ≈2sin⁡(θ/2)\sin \theta \approx 2 \sin(\theta/2). Substituting the approximation recursively, sin⁡(θ/2)≈θ/2\sin(\theta/2) \approx \theta/2, gives sin⁡θ≈2⋅(θ/2)=θ\sin \theta \approx 2 \cdot (\theta/2) = \theta. This process can be repeated by halving the angle multiple times, approaching the limit behavior algebraically without invoking infinite series. For the cosine function, the approximation cos⁡θ≈1−θ2/2\cos \theta \approx 1 - \theta^2/2 follows algebraically from the known sine limit and the identity 1−cos⁡θ=2sin⁡2(θ/2)1 - \cos \theta = 2 \sin^2(\theta/2). Substituting into the rearranged limit form, lim⁡θ→01−cos⁡θθ2=12\lim_{\theta \to 0} \frac{1 - \cos \theta}{\theta^2} = \frac{1}{2}, yields: 1−cos⁡θθ2=2sin⁡2(θ/2)θ2=12(sin⁡(θ/2)θ/2)2.\frac{1 - \cos \theta}{\theta^2} = \frac{2 \sin^2(\theta/2)}{\theta^2} = \frac{1}{2} \left( \frac{\sin(\theta/2)}{\theta/2} \right)^2. As θ→0\theta \to 0, sin⁡(θ/2)θ/2→1\frac{\sin(\theta/2)}{\theta/2} \to 1, confirming the limit value of 1/21/2 and thus the approximation. The tangent approximation tan⁡θ≈θ\tan \theta \approx \theta is derived algebraically by substituting the sine and cosine approximations into the definition tan⁡θ=sin⁡θ/cos⁡θ\tan \theta = \sin \theta / \cos \theta: tan⁡θ≈θ1−θ2/2≈θ(1+θ22)≈θ,\tan \theta \approx \frac{\theta}{1 - \theta^2/2} \approx \theta \left(1 + \frac{\theta^2}{2}\right) \approx \theta, where the higher-order term θ3/2\theta^3/2 is neglected for small θ\theta. This finite manipulation preserves the leading-order behavior without series expansion. Alternatively, these approximations can be obtained through finite polynomial fitting methods, such as least-squares approximation or interpolation, by matching polynomial coefficients to tabulated values of the trigonometric functions at small discrete points near θ=0\theta = 0. For instance, fitting a linear polynomial to sin⁡θ\sin \theta over points in [−ϵ,ϵ][-\epsilon, \epsilon] for small ϵ>0\epsilon > 0 yields the slope 1 and intercept 0, confirming sin⁡θ≈θ\sin \theta \approx \theta; similar fits for cos⁡θ\cos \theta and tan⁡θ\tan \theta produce the quadratic and linear forms, respectively. These algebraic techniques rely on solving linear systems from the least-squares criterion min⁡∑(f(θi)−p(θi))2\min \sum (f(\theta_i) - p(\theta_i))^2.

Error and Accuracy

Error Bounds

The absolute error in the small-angle approximation sin⁡θ≈θ\sin \theta \approx \theta (with θ\theta in radians) satisfies ∣sin⁡θ−θ∣≤θ36|\sin \theta - \theta| \leq \frac{\theta^3}{6} for small positive θ\theta, as established by the alternating series estimation theorem applied to the of sine. The relative error for this approximation is then approximately θ26\frac{\theta^2}{6}, since the true value sin⁡θ\sin \theta is close to θ\theta for small angles. For the approximation cos⁡θ≈1−θ22\cos \theta \approx 1 - \frac{\theta^2}{2}, the absolute error is bounded by θ424\frac{\theta^4}{24}, derived similarly from the remainder after the quadratic term. The corresponding relative error is approximately θ424\frac{\theta^4}{24}, given that cos⁡θ≈1\cos \theta \approx 1. For tan⁡θ≈θ\tan \theta \approx \theta, the leading error term from the is approximately θ33\frac{\theta^3}{3}, yielding a relative error of roughly θ23\frac{\theta^2}{3}. To illustrate these errors practically, consider the following table of relative errors (defined as ∣approximation−true valuetrue value∣×100%|\frac{\text{approximation} - \text{true value}}{\text{true value}}| \times 100\%) for the sin⁡θ≈θ\sin \theta \approx \theta approximation at selected small angles in radians:
θ\theta (radians)θ\theta (degrees)True sin⁡θ\sin \thetaRelative Error (%)
0.1≈5.7°0.0998330.17
0.5≈28.6°0.4794264.29
1.0≈57.3°0.84147118.85
These values demonstrate rapid growth in error as θ\theta increases; for instance, the relative error remains below 1% for θ≲0.2\theta \lesssim 0.2 radians (≈11.5°). In applications, the approximations are typically considered reliable to within 1% relative error for angles up to about 10° (≈0.175 radians), beyond which higher-order terms become significant and the linear or quadratic forms fail to capture the behavior accurately. For the tan⁡θ≈θ\tan \theta \approx \theta case specifically, the 1% threshold occurs near 10°, while for sin⁡θ≈θ\sin \theta \approx \theta it extends slightly further due to the smaller leading error term.

Higher-Order Terms

To achieve greater accuracy in the small-angle approximation for moderately small angles, higher-order terms from the expansions of the are incorporated. For the sine function, the approximation improves from sin⁡θ≈θ\sin \theta \approx \theta to sin⁡θ≈θ−θ36\sin \theta \approx \theta - \frac{\theta^3}{6}, where the cubic term corrects the leading-order error, leaving a remainder of order O(θ5)O(\theta^5). Similarly, for cosine, the refinement is cos⁡θ≈1−θ22+θ424\cos \theta \approx 1 - \frac{\theta^2}{2} + \frac{\theta^4}{24}, with the quartic term reducing the error to O(θ6)O(\theta^6). For , the higher-order form is tan⁡θ≈θ+θ33\tan \theta \approx \theta + \frac{\theta^3}{3}, truncating at the cubic term to achieve an error of O(θ5)O(\theta^5). These higher-order approximations are particularly useful for angles up to about 0.5 radians (approximately 28.6 degrees), where they substantially reduce the approximation error compared to the linear terms alone. The inclusion of the next significant term typically cuts the error by more than 90% for such angles, as the ratio of the leading neglected terms scales with θ2\theta^2, which is less than 0.25 at θ=0.5\theta = 0.5. For example, at θ=0.3\theta = 0.3 radians, the basic approximation sin⁡θ≈θ\sin \theta \approx \theta yields a relative error of approximately 1.5%, whereas the higher-order form sin⁡θ≈θ−θ36\sin \theta \approx \theta - \frac{\theta^3}{6} reduces this to about 0.007%, demonstrating an error reduction exceeding 99%. The accuracy of these truncated expansions can be rigorously bounded using the Lagrange form of the remainder in . For a function f(θ)f(\theta) expanded around 0 to order nn, the remainder after nn terms is Rn(θ)=f(n+1)(ξ)(n+1)!θn+1R_n(\theta) = \frac{f^{(n+1)}(\xi)}{(n+1)!} \theta^{n+1} for some ξ\xi between 0 and θ\theta. For the sine approximation up to the cubic term (n=3n=3), this becomes R3(θ)=sin⁡ξ24θ4R_3(\theta) = \frac{\sin \xi}{24} \theta^4, where ∣sin⁡ξ∣≤1|\sin \xi| \leq 1, so ∣R3(θ)∣≤θ424|R_3(\theta)| \leq \frac{\theta^4}{24}. This bound provides a quantitative limit for practical use, though the true error scales as O(θ5)O(\theta^5) due to the structure of the sine series.

Extensions

Angle Sum and Difference Approximations

The small-angle approximation extends naturally to sums and differences of angles through the trigonometric formulas, particularly when one is small relative to the other or when both are small. Consider the sum of angles θ and δ, where δ is small (in radians). Using the formula, sin⁡(θ+δ)=sin⁡θcos⁡δ+cos⁡θsin⁡δ,\sin(\theta + \delta) = \sin \theta \cos \delta + \cos \theta \sin \delta, and applying the basic approximations cos⁡δ≈1\cos \delta \approx 1 and sin⁡δ≈δ\sin \delta \approx \delta, this simplifies to sin⁡(θ+δ)≈sin⁡θ⋅1+cos⁡θ⋅δ=sin⁡θ+δcos⁡θ.\sin(\theta + \delta) \approx \sin \theta \cdot 1 + \cos \theta \cdot \delta = \sin \theta + \delta \cos \theta. A similar derivation for the cosine function yields cos⁡(θ+δ)=cos⁡θcos⁡δ−sin⁡θsin⁡δ≈cos⁡θ−δsin⁡θ.\cos(\theta + \delta) = \cos \theta \cos \delta - \sin \theta \sin \delta \approx \cos \theta - \delta \sin \theta. These forms are first-order approximations derived from the limit definitions underlying calculus, where the change in the function over a small increment δ approximates the derivative times δ. For the tangent function, the addition formula is tan⁡(θ+δ)=tan⁡θ+tan⁡δ1−tan⁡θtan⁡δ.\tan(\theta + \delta) = \frac{\tan \theta + \tan \delta}{1 - \tan \theta \tan \delta}. When both θ and δ are small, tan⁡θ≈θ\tan \theta \approx \theta and tan⁡δ≈δ\tan \delta \approx \delta, and the product tan⁡θtan⁡δ\tan \theta \tan \delta is second-order small, so it can be neglected, leading to tan⁡(θ+δ)≈θ+δ\tan(\theta + \delta) \approx \theta + \delta. More generally, if only δ is small, the approximation becomes tan⁡(θ+δ)≈tan⁡θ+δsec⁡2θ\tan(\theta + \delta) \approx \tan \theta + \delta \sec^2 \theta, which aligns with the differential d(tan⁡θ)=sec⁡2θ dθd(\tan \theta) = \sec^2 \theta \, d\theta. These extensions are useful in contexts requiring incremental changes, such as in differential equations or perturbation analysis. In applications involving differentials, the small-angle approximations manifest directly as d(sin⁡θ)≈cos⁡θ dθd(\sin \theta) \approx \cos \theta \, d\theta and d(cos⁡θ)≈−sin⁡θ dθd(\cos \theta) \approx -\sin \theta \, d\theta, providing linear approximations for changes in angle. This is foundational in for deriving rates of change in trigonometric expressions. Regarding errors, when approximating sums or differences of independent small angles, the uncertainties such that the variance of the total angle is the sum of the individual variances, meaning errors add quadratically: if angles α and β have uncertainties Δα and Δβ, then the uncertainty in α ± β is (Δα)2+(Δβ)2\sqrt{(\Delta \alpha)^2 + (\Delta \beta)^2}
Add your contribution
Related Hubs
User Avatar
No comments yet.