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The small-angle approximations can be used to approximate the values of the main
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
, provided that the angle in question is small and is measured in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s: : \begin \sin \theta &\approx \theta \\ \cos \theta &\approx 1 - \frac \approx 1\\ \tan \theta &\approx \theta \end These approximations have a wide range of uses in branches of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, including
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
,
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
,
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
,
cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an im ...
,
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, and
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
. One reason for this is that they can greatly simplify
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s 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 Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ( ...
for each of the trigonometric functions. Depending on the order of the approximation, \textstyle \cos \theta is approximated as either 1 or as 1-\frac.


Justifications


Graphic

The accuracy of the approximations can be seen below in Figure 1 and Figure 2. As the measure of the angle approaches zero, the difference between the approximation and the original function also approaches 0. File:Small_angle_compair_odd.svg, Figure 1. A comparison of the basic
odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
trigonometric functions to . It is seen that as the angle approaches 0 the approximations become better. File:Small_angle_compare_even.svg, Figure 2. A comparison of to . It is seen that as the angle approaches 0 the approximation becomes better.


Geometric

The red section on the right, , is the difference between the lengths of the hypotenuse, , and the adjacent side, . As is shown, and are almost the same length, meaning is close to 1 and helps trim the red away. \cos \approx 1 - \frac The opposite leg, , is approximately equal to the length of the blue arc, . Gathering facts from geometry, , from trigonometry, and , and from the picture, and leads to: \sin \theta = \frac\approx\frac = \tan \theta = \frac \approx \frac = \frac = \theta. Simplifying leaves, \sin \theta \approx \tan \theta \approx \theta.


Calculus

Using the
squeeze theorem In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is trapped between two other functions. The squeeze theorem is used in calculus and mathematical anal ...
, we can prove that \lim_ \frac = 1, which is a formal restatement of the approximation \sin(\theta) \approx \theta for small values of ''θ''. A more careful application of the squeeze theorem proves that \lim_ \frac = 1, from which we conclude that \tan(\theta) \approx \theta for small values of ''θ''. Finally,
L'Hôpital's rule In calculus, l'Hôpital's rule or l'Hospital's rule (, , ), also known as Bernoulli's rule, is a theorem which provides a technique to evaluate limits of indeterminate forms. Application (or repeated application) of the rule often converts an i ...
tells us that \lim_ \frac = \lim_ \frac = -\frac, which rearranges to \cos(\theta) \approx 1 - \frac for small values of ''θ''. Alternatively, we can use the
double angle formula In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
\cos 2A \equiv 1-2\sin^2 A. By letting \theta = 2A, we get that \cos\theta=1-2\sin^2\frac\approx1-\frac.


Algebraic

The Maclaurin expansion (the Taylor expansion about 0) of the relevant trigonometric function is \begin \sin \theta &= \sum^_ \frac \theta^ \\ &= \theta - \frac + \frac - \frac + \cdots \end where is the angle in radians. In clearer terms, \sin \theta = \theta - \frac + \frac - \frac + \cdots It is readily seen that the second most significant (third-order) term falls off as the cube of the first term; thus, even for a not-so-small argument such as 0.01, the value of the second most significant term is on the order of , or the first term. One can thus safely approximate: \sin \theta \approx \theta By extension, since the cosine of a small angle is very nearly 1, and the tangent is given by the sine divided by the cosine, \tan \theta \approx \sin \theta \approx \theta,


Error of the approximations

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 0.1408 radians (8.07°) * at about 0.1730 radians (9.91°) * at about 0.2441 radians (13.99°) * at about 0.6620 radians (37.93°)


Angle sum and difference

The
angle addition and subtraction theorems In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
reduce to the following when one of the angles is small (''β'' ≈ 0): :


Specific uses


Astronomy

In
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, the
angular size The angular diameter, angular size, apparent diameter, or apparent size is an angular distance describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the visual angle, and in optics, it is ...
or angle subtended by the image of a distant object is often only a few
arcsecond A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The na ...
s, so it is well suited to the small angle approximation. The linear size () is related to the angular size () and the distance from the observer () by the simple formula: :D = X \frac where is measured in arcseconds. The number is approximately equal to the number of arcseconds in a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
(), divided by . The exact formula is :D = d \tan \left( X \frac \right) and the above approximation follows when is replaced by .


Motion of a pendulum

The second-order cosine approximation is especially useful in calculating the
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
of a
pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...
, which can then be applied with a
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
to find the indirect (energy) equation of motion. When calculating the
period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (or rhetorical period), a concept ...
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 In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
.


Optics

In optics, the small-angle approximations form the basis of the
paraxial approximation In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). A paraxial ray is a ray which makes a small angle (''θ'') to the optical ...
.


Wave Interference

The sine and tangent small-angle approximations are used in relation to the
double-slit experiment In modern physics, the double-slit experiment is a demonstration that light and matter can display characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechanics ...
or a
diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form of structura ...
to simplify equations, e.g. 'fringe spacing' = 'wavelength' × 'distance from slits to screen' ÷ 'slit separation'.


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 In structural engineering, buckling is the sudden change in shape (deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a gr ...
). 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 The basic principles of air navigation are identical to general navigation, which includes the process of planning, recording, and controlling the movement of a craft from one place to another. Successful air navigation involves piloting an air ...
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 In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tables w ...
values: Example: sin(0.755) \begin \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 where the values for sin(0.75) and cos(0.75) are obtained from trigonometric table


See also

*
Skinny triangle In trigonometry, a skinny triangle is a triangle whose height is much greater than its base. The solution of triangles, solution of such triangles can be greatly simplified by using the approximation that the sine of a small angle is equal to that ...
* Infinitesimal oscillations of a pendulum *
Versine and haversine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',Exsecant and excosecant The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric functions defined in terms of the secant and cosecant functions. They used to be important in fields such as surveying, railway engineering, civil engineering, astro ...


References

{{DEFAULTSORT:Small-Angle Approximation Trigonometry Equations of astronomy