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Simplification of the basic trigonometric functions
small angles, the trigonometric functions sine, cosine, and tangent can be calculated with reasonable accuracy by the following simple approximations:
Small-angle_approximation
Small angle approximation in geometric optics
In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system
Paraxial_approximation
Free swinging suspended body
centre-of-mass θ {\displaystyle \theta } is the angle from the vertical Hence, under the small-angle approximation, sin θ ≈ θ {\displaystyle \sin \theta \approx
Pendulum_(mechanics)
To-and-fro periodic motion in science and engineering
displacement (and even so, it is only a good approximation when the angle of the swing is small; see small-angle approximation). Simple harmonic motion can also
Simple_harmonic_motion
Rule of thumb used in aviation
separation purposes. The rule is based on the small-angle approximation (which states that, for small angles, sin θ ≈ θ, where θ is in radians), along with
1_in_60_rule
Type of triangle
triangles can be greatly simplified by using the approximation that the sine of a small angle is equal to that angle in radians. The solution is particularly
Skinny_triangle
Something roughly the same as something else
to represent a quantity Small-angle approximation – Simplification of the basic trigonometric functions Successive-approximation ADC – Type of analog-to-digital
Approximation
Type of wave
{\partial ^{2}y}{\partial t^{2}}}.} According to the small-angle approximation, the tangents of the angles at the ends of the string piece are equal to the
String_vibration
Measure of relativistic velocity
{v}{c}})} where c is the speed of light. For low speeds, by the small-angle approximation, w is approximately v / c. Since in relativity any velocity v
Rapidity
System where changes of output are not proportional to changes of input
involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that | θ | {\displaystyle |\theta
Nonlinear_system
Phase of a cycle
Consider a planar pendulum with constant natural frequency ω in the small angle approximation. There are two forces acting on the pendulum bob: the restoring
Geometric_phase
Mathematical relation in spherical triangles
c^{2}\approx a^{2}+b^{2}-2ab\cos C\,.} To prove this, we will use the small-angle approximation obtained from the Maclaurin series for the cosine and sine functions:
Spherical_law_of_cosines
Shortest distance between two points on the surface of a sphere
short-distance approximation ( | Δ σ c | {\displaystyle |\Delta \sigma _{\text{c}}|} much smaller than 1 {\displaystyle 1} , cf. Small-angle approximation), Δ σ
Great-circle_distance
Angular measurement, thousandth of a radian
Milliradians are generally used for very small angles, which allows for very accurate mathematical approximations to more easily calculate with direct proportions
Milliradian
other quantum phenomena. Though a pendulum not subject to the small-angle approximation has an inherent nonlinearity, the Schrödinger equation for the
Quantum_pendulum
Concept in geometry
sin(d𝜃) = 1/2 · r2 · d𝜃). Note that sin(d𝜃) ≈ d𝜃 due to small angle approximation. Through summing the areas of the triangles, the expression for
Area_of_a_circle
Clarity of vision
{S^{2}}{4}}}},} where the S2/4 term is omitted for small objects relative to great distances by small-angle approximation. To exceed the SAVT, an object of size S
Visual_acuity
Analyzing a sample by studying trajectories of radiation passing through it
Small-angle scattering (SAS) is a scattering technique based on deflection of collimated radiation away from the straight trajectory after it interacts
Small-angle_scattering
Extent of the observable world seen at any given moment
linear field of view in millimeters per meter. Then, using the small-angle approximation: A ≈ 360 ∘ 2 π ⋅ M 1000 ≈ 0.0573 × M {\displaystyle A\approx {360^{\circ
Field_of_view
Mathematical approximation of a function
light rays making small angles with an axis is studied by replacing the sine function with its linear approximation. Such approximations are used throughout
Taylor_series
Area of geometry, about angles and lengths
trigonometric identities Rational trigonometry Skinny triangle Small-angle approximation Trigonometric functions Unit circle Uses of trigonometry Harper
Trigonometry
Mathematical function in general imaging
shown to be proportional to the cube of the incident angle (θ) under the small-angle approximation, and that the explicit form in this case is r s = C
Contrast_transfer_function
Distance over which a wave's shape repeats
small difference. Consequently, interference occurs. In the Fraunhofer diffraction pattern sufficiently far from a single slit, within a small-angle approximation
Wavelength
Branch of physics that studies light
Geometric optics is often simplified by making the paraxial approximation, or "small angle approximation". The mathematical behaviour then becomes linear, allowing
Optics
Angle between the two sightlines or two objects as viewed from an observer
we can develop the above expression and simplify it. In the small-angle approximation, at second order, the above expression becomes: cos θ ≈ 1 −
Angular_distance
Laws in physics about force and motion
{\displaystyle \theta } is nearly equal to θ {\displaystyle \theta } (see small-angle approximation), and so this expression simplifies to the equation for a simple
Newton's_laws_of_motion
Streamlined body for generating lift
w(x)} must balance an inverse flow from V. By the small-angle approximation, V is inclined at angle α-dy⁄dx relative to the blade at position x, and the
Airfoil
Optical device which transmits and refracts light
\theta &={\frac {h}{R}}\end{aligned}}} , and using small angle approximation (paraxial approximation) and eliminating i, r, and θ, n 2 v + n 1 u = n 2
Lens
Mathematical model to quantify lift
across a wing may still have varying α(y)). By the small-angle approximation, the effective angle of attack at y of the combined freestream and vortex
Lifting-line_theory
Functions of an angle
complementary angle) and proceeds to define the cotangens similarly. Bhāskara I's sine approximation formula Small-angle approximation Differentiation
Trigonometric_functions
Tensor in general relativity
\phi } direction. Using some elementary trigonometry and the small angle approximation, we find that the force vectors differ by a vector tangent to
Tidal_tensor
Electrical circuit detecting phase difference
using some approximations. Assume that the phase differences will be small (much less than 1 radian, for example). The small-angle approximation for the
Phase_detector
Numerical method used in computational fluid dynamics
aerodynamic forces are neglected. The angle of attack and the angle of sideslip are both small, small angle approximation. By the above assumptions the flowfield
Vortex_lattice_method
Concept in railway engineering
\theta }}.} Geometrically, tan θ can be expressed (using the small-angle approximation) in terms of the track gauge G, the cant ha and cant deficiency
Minimum_railway_curve_radius
Flow behind a conical shock wave
and consequently g ′ = U χ {\displaystyle g'=U\chi } . In the small-angle approximation, the weak shock cone is given by z = β ρ {\displaystyle z=\beta
Taylor–Maccoll_flow
Set of bivariate correlations among galaxies
determine the distance of the galaxy from the observer through the small-angle approximation. An early use of the fundamental plane is the D n − σ o {\displaystyle
Fundamental plane (elliptical galaxies)
Fundamental_plane_(elliptical_galaxies)
Spacetime modeled by four pointwise-orthonormal vector fields
{\displaystyle r=r_{0}} . Using some elementary trigonometry and the small angle approximation, we find that the force vectors differ by a vector tangent to
Frame fields in general relativity
Frame_fields_in_general_relativity
Method for structure analysis of biological materials
Biological small-angle scattering is a small-angle scattering method for structure analysis of biological materials. Small-angle scattering is used to
Biological small-angle scattering
Biological_small-angle_scattering
Study of classical optics using Fourier transforms
\over c}} is the wave number. Next, use the paraxial approximation, that is a small-angle approximation such that k x 2 + k y 2 ≪ k z 2 {\displaystyle
Fourier_optics
Model of optics describing light as geometric rays
Geometrical optics is often simplified by making the paraxial approximation, or "small angle approximation". The mathematical behavior then becomes linear, allowing
Geometrical_optics
Approximation of a function by its tangent line at a point
paraxial approximation, in which only rays which make small angles with the optical axis of the system are considered. In this approximation, trigonometric
Linear_approximation
Phenomenon in which a neutrino changes lepton flavor as it travels
not identical the analysis is slightly more complicated. In the small-angle approximation, the potential energy of a single pendulum system is 1 2 m g L
Neutrino_oscillation
Scattering theory
reflection in a neutron guide, or grazing-incidence small-angle scattering. Using the first Born approximation, it has been shown that the scattering amplitude
Born_approximation
Mechanism for regulating the speed of clocks
difference between the true period and the small angle approximation (1) amounts to about 15 seconds per day. For small swings the pendulum approximates a harmonic
Pendulum
Safe driving distance between cars
so in such conditions, a fieldable small angle approximation may be used. Substituting small-angle approximations sin θ ≈ θ, cos ≈ 1 − θ2/2: v ≈ g r [
Assured_clear_distance_ahead
Quantum mechanics thought experiment
{\textstyle \sin ^{2}(\epsilon )\approx \epsilon ^{2}} by the small-angle approximation. Otherwise, the qubit will collapse to |0⟩ and the circuit will
Elitzur–Vaidman_bomb_tester
Figure formed by two rays meeting at a common point
form a right angle are said to be normal, orthogonal, or perpendicular. An angle larger than a right angle and smaller than a straight angle (between 90°
Angle
Geometric object
\end{aligned}}} Note that this formula is only accurate under the small angle approximation. While the doublet prism is the simplest compound prism type,
Compound_prism
Ability of any image-forming device to distinguish small details of an object
separation. Using a small-angle approximation, the angular resolution may be converted into a spatial resolution, Δℓ, by multiplication of the angle (in radians)
Angular_resolution
Force resisting the motion when a body rolls on a surface
which has the dimension of length, is approximately (due to the small-angle approximation of cos ( θ ) = 1 {\displaystyle \cos(\theta )=1} ) equal to
Rolling_resistance
Changed position of star vs background
{\displaystyle d} is the distance to the star. Using small-angle approximations (valid when the angle is small compared to 1 radian), tan x ≈ x radians =
Stellar_parallax
Mechanism for regulating the speed of clocks
the pendulum is only approximately linear in the regime of the small angle approximation. To be time-independent, the path must be cycloidal. To minimize
Escapement
Reversible free swinging pendulum
acceleration of gravity g because for narrow swings (when the small angle approximation is valid) its period of swing T depends only on g and its length
Kater's_pendulum
form factors as in conventional small-angle scattering (SAS). The standard scattering theory, first-order Born approximation must be extended to include two
Grazing-incidence small-angle scattering
Grazing-incidence_small-angle_scattering
}{x}}\quad &{\text{if }}x<0.\end{cases}}\end{aligned}}} By the small-angle approximation, both cos x {\displaystyle \cos {\sqrt {x}}} and sinc x {\displaystyle
Stumpff_function
Interference pattern
{\alpha }{2}}}}.\end{aligned}}} When α is very small (α < π/6) the following small-angle approximations can be made: sin α ≈ α cos α ≈ 1 {\displaystyle
Moiré_pattern
Images of gravitational lensing
(2008). "THEORETICAL GRAVITATIONAL LENSING – BEYOND THE WEAK-FIELD SMALL-ANGLE APPROXIMATION". The Eleventh Marcel Grossmann Meeting. World Scientific Publishing
Relativistic_images
Extraterrestrial view of outer space
only for spherical objects that are relatively close, since the small-angle approximation holds for x ≪ 1 {\displaystyle x\ll 1} : arcsin x ≈ arctan
Extraterrestrial_sky
Self-similar curve related to golden ratio
true logarithmic spiral, closely approximates a golden spiral. Another approximation is a Fibonacci spiral, which is constructed slightly differently. A
Golden_spiral
Telescope for observations with visible light
For visible light ( λ {\displaystyle \lambda } = 550 nm) in the small-angle approximation, this equation can be rewritten: α R = 138 D {\displaystyle \alpha
Optical_telescope
Angle between a liquid–vapor interface and a solid surface
The contact angle (symbol θC) is the angle between a liquid surface and a solid surface where they meet. More specifically, it is the angle between the
Contact_angle
1736 Leonhard Euler writes on the pendulum equation without the small-angle approximation. 1738 Euler writes on curves of genus 1 considered by Fermat and
Timeline_of_abelian_varieties
Application of mechanical dynamics to model the flight of space vehicles
velocity before plane change and φ is the flight path angle. Using the small-angle approximation, this becomes: Δ v = v cos ( φ ) | Δ i | {\displaystyle
Spacecraft_flight_dynamics
Mathematical symbol for "greater than"
sometimes used for an approximation of the closing angle bracket, ⟩. The proper Unicode character is U+232A 〉 RIGHT-POINTING ANGLE BRACKET. ASCII does not
Greater-than_sign
Mathematical symbol for "less than"
symbols: The less-than sign may be seen for an approximation of the opening angle bracket, ⟨. True angle bracket characters, as required in linguistics
Less-than_sign
Lists of values of mathematical functions
combine a polynomial or rational approximation (such as Chebyshev approximation, best uniform approximation, Padé approximation, and typically for higher or
Trigonometric_table
How large a sphere or circle appears
objects of large angular diameter, since the following small-angle approximations hold for small values of x {\displaystyle x} : arcsin x ≈ arctan
Angular_diameter
Process by which dust, particulates, etc. scatter light
properties in DDA approximation. Rayleigh scattering regime is the scattering of light, or other electromagnetic radiation, by particles much smaller than the
Light_scattering_by_particles
Theorem in physics
integrate over a screen far away in the xy plane, which is small enough for the small-angle approximations to be appropriate, but large enough that we can integrate
Optical_theorem
Set of quantities in accelerator physics
Given this assumption, it is possible to use the small angle approximation to express x' and y' as angles rather than simple ratios. As such, x' and y' are
Courant–Snyder_parameters
Approximation of a function by a polynomial
In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree
Taylor's_theorem
Substituting rare characters with more common characters
A typographic approximation is a replacement of an element of the writing system (usually a glyph) with another glyph or glyphs. The replacement may be
Typographic_approximation
Construction of an angle equal to one third a given angle
Angle trisection is the construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass
Angle_trisection
under an exit angle αf that is close to the critical angle for total external reflection αc. For x-rays and neutrons, critical angles are small, and the Yoneda
Yoneda_peak
Idealized model of light
simplification is useful in practice; it is an excellent approximation when the wavelength is small compared to the size of structures with which the light
Ray_(optics)
Lens with a thickness that is negligible
2 {\displaystyle r_{2}} is the angle of incidence and e {\displaystyle e} is the angle of refraction. For small angles, sin x ≈ x {\textstyle \sin x\approx
Thin_lens
Image edge detection algorithm
Technically, it is a discrete differentiation operator, computing an approximation of the gradient of the image intensity function. At each point in the
Sobel_operator
Shape with seven sides
γωνἰα, romanized: gonía, meaning angle. A regular heptagon, in which all sides and all angles are equal, has internal angles of 5 7 π {\displaystyle {\tfrac
Heptagon
Apparent curve that separates earth from sky
applied as a fairly good approximation when atmospheric conditions are close to standard. When conditions are unusual, this approximation fails. Refraction is
Horizon
Geometric description of eye positioning and movement
gaze direction has equation x = 0 {\displaystyle x=0} . Using small-angle approximation, the Fick system implies that near the origin, the velocity surface
Listing's_law
Type of air data system
The exact computation of the synthetic airspeed can vary (e.g., small-angle approximation can be made to simplify the computation), but it is primarily
Synthetic_air_data_system
Solution method for linear differential equations
In mathematical physics, the WKB approximation or WKB method is a technique for finding approximate solutions to linear differential equations with spatially
WKB_approximation
Mathematical book authored by Galileo Galilei
In other words, Galileo argues that his assumption relies on a small-angle approximation. In the defense of his assumption, Galileo states, “To such objectors
De_motu_antiquiora
Light scattering by small particles
the discrete dipole approximation and other computational techniques. Rayleigh scattering applies to particles that are small with respect to wavelengths
Rayleigh_scattering
Scientific method to study solutions, gels, and other polymeric systems
transition utilises once again the small-angle approximation. We can thus approximate the scattering intensity in the small-angle regime as: log I ( k ) = −
Polymer_scattering
π / λ {\displaystyle k=2\pi /\lambda } ). In the case of the small-angle approximation ( θ ∼ {\displaystyle \theta \sim } 100 mRad) we can approximate
Multislice
Near-field diffraction
the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff–Fresnel diffraction that can be applied to the propagation
Fresnel_diffraction
Characteristic of an optical system
between the numerical aperture and the acceptance angle of the fiber becomes only an approximation. In particular, "NA" defined this way is not relevant
Numerical_aperture
location on the globe is a direct effect of Sun angle on climate, as the solar zenith angle (the angle of incidence at which sunlight strikes Earth) varies
Effect of Sun angle on climate
Effect_of_Sun_angle_on_climate
Unit of measurement of an angle, equal to 1/400th of a circle
Greek γωνία (gōnía) 'angle'), grad, or grade – is a unit of measurement of an angle, defined as one-hundredth of the right angle; in other words, 100
Gradian
Rocket engine nozzle shape
slope so that at the nozzle exit the divergence angle is small, usually less than a 10 degree half angle. An ideal nozzle would direct all of the gases
Bell_nozzle
Area bounded by a circular arc and a straight line
{c}{16h}}(c^{2}-4h^{2})} What can be stated is that as the central angle gets smaller (or alternately the radius gets larger), the area a rapidly and asymptotically
Circular_segment
similarly described by this differential equation and solution in the small angle approximation. We can use the binomial expansion to rewrite ω {\displaystyle
Cantilever_magnetometry
Energy transfer in the form of electromagnetic radiation
Eddington approximation is distinct from the two-stream approximation. The two-stream approximation assumes that the intensity is constant with angle in the
Radiative_transfer
symbol column is either an image, an html creation, or a typographical approximation using a similar unicode symbol. Depending on local conventions, selected
List_of_proofreader's_marks
Technique in geometric optics
paraxial approximation, in which only rays that make small angles with the optical axis of the system are considered. In this approximation, the trigonometric
Gaussian_optics
Varying methods used to calculate pi
Approximations for the mathematical constant pi (π) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning
Approximations_of_pi
SI derived unit of angle
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
Radian
Periodic motion of the atoms of a molecule
To a first approximation, the motion in a normal vibration can be described as a kind of simple harmonic motion. In this approximation, the vibrational
Molecular_vibration
Diffraction pattern in optics
definite radius. If two objects imaged by a camera are separated by an angle small enough that their Airy disks on the camera detector start overlapping
Airy_disk
SMALL ANGLE-APPROXIMATION
SMALL ANGLE-APPROXIMATION
Female
English
English short form of Latin Angela, ANGIE means "angel, messenger."
Male
French
French name ANGE means "angel, messenger." Compare with feminine Ange.
Boy/Male
Spanish American Greek Latin
Angel.
Boy/Male
American, Danish, French, German, Greek, Indian, Italian, Spanish
Angel
Girl/Female
French
Angel.
Surname or Lastname
English
English : nickname for a person of slender build or diminutive stature, from Middle English smal ‘thin’, ‘narrow’.Translation of equivalents in other European languages, such as German Klein and Schmal, French Petit.
Girl/Female
Greek American Italian Latin
Messenger or angel. A popular masculine name in Sicily after the 13th-century saint, Angel. Angel...
Girl/Female
Christian, French, German, Greek
Angel; A Messenger from God
Surname or Lastname
English
English : patronymic from Small.
Male
English
English unisex name derived from Latin Angelus, ANGEL means "angel, messenger." Once used as a man's name in England. It is now almost strictly a feminine name.
Surname or Lastname
English (Devon)
English (Devon) : variant of Small.
Girl/Female
Indian
Pari fairy
Boy/Male
American, British, Danish, English, French, German, Greek, Hindu, Indian, Latin, Spanish
Messenger of God; Angel; Messenger
Girl/Female
Latin American Greek
Angel; Like an angel. From angelicus meaning angelic.
Girl/Female
African, Arabic, Australian, Celebrity, Danish, German, Gujarati, Hindu, Indian, Kannada, Marathi, Muslim, Swahili
Angel; Small Child
Female
English
English short form of Latin Angela, ANGE means "angel, messenger." Compare with masculine Ange.
Boy/Male
German, Swedish
Angel; Bright Angle
Surname or Lastname
English and Irish (of Norman origin)
English and Irish (of Norman origin) : topographic name from Middle English and Old French angle ‘angle’, ‘corner’ (Latin angulus). As an Irish surname, it can also be habitational, from a place in Pembrokeshire, South Wales, named with this word.Americanized spelling of German Angel or Engel.
Girl/Female
British, English, Greek, Latin
Angel
Female
English
English unisex name derived from Latin Angelus, ANGEL means "angel, messenger."Â Originally a male name, it is now almost strictly female.
SMALL ANGLE-APPROXIMATION
SMALL ANGLE-APPROXIMATION
Surname or Lastname
English (Somerset)
English (Somerset) : possibly from the Welsh patronymic ap Ridel ‘son of Ridel’.
Girl/Female
Tamil
Nicika | நீஸீகா  Â
Perfect
Boy/Male
Tamil
Girl/Female
Hindu, Indian, Marathi, Tamil
The Earth
Girl/Female
Indian
No One else
Girl/Female
Hindu
A flower
Girl/Female
German
Whole. Complete. Universal. Old German 'ermin' or 'irmin', meaning universal. Also a feminine...
Girl/Female
Indian
Ornament, Abbreviation of names ending in -gina
Girl/Female
Hindi American
Black.
Male
French
Norman French form of Middle English Gawain, GAUVAIN means either "May hawk" or "white hawk."
SMALL ANGLE-APPROXIMATION
SMALL ANGLE-APPROXIMATION
SMALL ANGLE-APPROXIMATION
SMALL ANGLE-APPROXIMATION
SMALL ANGLE-APPROXIMATION
a.
Relating to an angle or to angles; having an angle or angles; forming an angle or corner; sharp-cornered; pointed; as, an angular figure.
a.
Having eight angles; eight-angled.
adv.
In or to small extent, quantity, or degree; little; slightly.
a.
Having an angle or angles; -- used in compounds; as, right-angled, many-angled, etc.
n.
The small or slender part of a thing; as, the small of the leg or of the back.
a.
Having oblique angles; as, an oblique-angled triangle.
imp. & p. p.
of Angle
v. i.
To use some bait or artifice; to intrigue; to scheme; as, to angle for praise.
v. t.
To cajole or coax; to wheedle. See Engle.
n.
A favorite; a paramour; an ingle.
a.
Having acute angles; as, an acute-angled triangle, a triangle with every one of its angles less than a right angle.
n.
The difference of direction of two lines. In the lines meet, the point of meeting is the vertex of the angle.
superl.
Being of slight consequence; feeble in influence or importance; unimportant; trivial; insignificant; as, a small fault; a small business.
n.
See Ankle.
n.
One who angles.
n.
A paramour; a favourite; a sweetheart; an engle.
v. i.
To fish with an angle (fishhook), or with hook and line.
v. t.
To cause to dangle; to swing, as something suspended loosely; as, to dangle the feet.
a.
Containing a right angle or right angles; as, a right-angled triangle.