language:
Find link is a tool written by Edward Betts.Longer titles found: Sine and cosine transforms (view), Fourier sine and cosine series (view)
searching for Sine and cosine 166 found (212 total)
alternate case: sine and cosine
Pythagorean trigonometric identity
(2,083 words)
[view diff]
exact match in snippet
view article
find links to article
sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions. The identity is sin 2 θ + cos 2 θ = 1. {\displaystylePure tone (615 words) [view diff] exact match in snippet view article find links to article
amplitude change between such a system's pure-tone input and its output. Sine and cosine waves can be used as basic building blocks of more complex waves. AsEmpirical orthogonal functions (343 words) [view diff] exact match in snippet view article find links to article
analysis typically uses predetermined orthogonal functions, for example, sine and cosine functions at fixed frequencies. In some cases the two methods may yieldList of integrals of hyperbolic functions (2,004 words) [view diff] no match in snippet view article find links to article
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integralsGolden angle (692 words) [view diff] exact match in snippet view article find links to article
follow from well-known algebraic properties of the golden ratio. As its sine and cosine are transcendental numbers, the golden angle cannot be constructedSine wave (1,332 words) [view diff] exact match in snippet view article find links to article
combination of two sine waves with phases of zero and a quarter cycle, the sine and cosine components, respectively. Sine wave Five seconds of a 220 Hz sine waveMadhava series (4,624 words) [view diff] exact match in snippet view article find links to article
later independently discovered in 17th century Europe. The series for sine and cosine were rediscovered by Isaac Newton in 1669, and the series for arctangentYuktibhāṣā (1,996 words) [view diff] exact match in snippet view article find links to article
Taylor series, including second and third order approximations of sine and cosine; radii, diameters and circumferences. Yuktibhāṣā mainly gives rationaleTangent half-angle substitution (3,042 words) [view diff] exact match in snippet view article find links to article
ISBN 978-1-108-41881-2. Other trigonometric functions can be written in terms of sine and cosine. Gunter, Edmund (1673) [1624]. The Works of Edmund Gunter. FrancisTimeline of geometry (1,413 words) [view diff] exact match in snippet view article find links to article
numerical values. It defines the concepts of sine and cosine, and also contains the earliest tables of sine and cosine values (in 3.75-degree intervals fromTranscendental function (1,945 words) [view diff] exact match in snippet view article find links to article
functions sine and cosine by introducing (−1)k into the series, resulting in alternating series. After Euler, mathematicians view the sine and cosine thisSimple cell (912 words) [view diff] exact match in snippet view article find links to article
when seeing an isolated bar. Using the mathematical Gabor model with sine and cosine components (phases), complex cells are then modeled by computing theTrigonometric functions of matrices (601 words) [view diff] exact match in snippet view article find links to article
The trigonometric functions (especially sine and cosine) for complex square matrices occur in solutions of second-order systems of differential equationsResolver (electrical) (980 words) [view diff] exact match in snippet view article
fixed at right (90°) angles to each other on the stator, produce a sine and cosine feedback current. The relative magnitudes of the two-phase voltagesEntire function (3,277 words) [view diff] exact match in snippet view article find links to article
products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivativesEuler's formula (3,851 words) [view diff] exact match in snippet view article find links to article
between analysis and trigonometry, and provides an interpretation of the sine and cosine functions as weighted sums of the exponential function: cos x = ReDe Moivre's formula (2,449 words) [view diff] exact match in snippet view article find links to article
complex numbers z such that zn = 1. Using the standard extensions of the sine and cosine functions to complex numbers, the formula is valid even when x is anLock-in amplifier (2,185 words) [view diff] exact match in snippet view article find links to article
by fast digital signal processing, for example, on an FPGA. Usually sine and cosine demodulation is performed simultaneously, which is sometimes also referredDixon elliptic functions (5,056 words) [view diff] exact match in snippet view article find links to article
{\displaystyle x^{3}+y^{3}=1} , just as the trigonometric functions sine and cosine parametrize the unit circle x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1}Proofs of trigonometric identities (5,714 words) [view diff] exact match in snippet view article find links to article
definitions, and therefore other proofs are based on the Taylor series of sine and cosine, or on the differential equation f ″ + f = 0 {\displaystyle f''+f=0}Indian mathematics (14,795 words) [view diff] exact match in snippet view article find links to article
further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmittedPeriodic function (1,704 words) [view diff] exact match in snippet view article find links to article
{\displaystyle f} is the sawtooth wave. The trigonometric functions sine and cosine are common periodic functions, with period 2 π {\displaystyle 2\piComplex number (11,623 words) [view diff] exact match in snippet view article find links to article
} (These are a consequence of the trigonometric identities for the sine and cosine function.) In other words, the absolute values are multiplied and theAP Precalculus (457 words) [view diff] case mismatch in snippet view article find links to article
Phenomena 2 3.2 Sine, Cosine, and Tangent 2 3.3 Sine and Cosine Function Values 2 3.4 Sine and Cosine Function Graphs 2 3.5 Sinusoidal Functions 2 3.6Moyal bracket (1,359 words) [view diff] exact match in snippet view article find links to article
differentiable phase-space functions, and f g is the ordinary product. The sine and cosine brackets are, respectively, the results of antisymmetrizing and symmetrizing1571 in science (197 words) [view diff] exact match in snippet view article find links to article
singularis containing many trigonometric tables and formulas on the sine and cosine, and novel in using a decimal notation; publication continued untilLeast-squares spectral analysis (3,354 words) [view diff] exact match in snippet view article find links to article
simplification in general, except for pair-wise correlations between sine and cosine bases of the same frequency, since the correlations between pairs ofBernoulli polynomials (4,252 words) [view diff] exact match in snippet view article find links to article
limit of large degree, they approach, when appropriately scaled, the sine and cosine functions. A similar set of polynomials, based on a generating functionA History of the Kerala School of Hindu Astronomy (711 words) [view diff] exact match in snippet view article find links to article
Newton-Gauss interpolation formula, Taylor series for sine and cosine functions, power series for sine and cosine functions, Lhuier's formula for the circum-radiusPhase-shift keying (6,226 words) [view diff] exact match in snippet view article find links to article
frequency carrier wave. The modulation is accomplished by varying the sine and cosine inputs at a precise time. It is widely used for wireless LANs, RFIDBivariate von Mises distribution (697 words) [view diff] exact match in snippet view article find links to article
commonly used variants of the bivariate von Mises distribution are the sine and cosine variant. The cosine variant of the bivariate von Mises distributionUnit interval (759 words) [view diff] exact match in snippet view article find links to article
occurs frequently, such as in the range of the trigonometric functions sine and cosine and the hyperbolic function tanh. This interval may be used for theKerala school of astronomy and mathematics (3,516 words) [view diff] exact match in snippet view article find links to article
given in two papers, a commentary on the Yuktibhasa's proof of the sine and cosine series and two papers that provide the Sanskrit verses of the TantrasangrahavakhyaCadambathur Tiruvenkatacharlu Rajagopal (510 words) [view diff] exact match in snippet view article find links to article
Scripta Mathematica 17: 65–74. 1949: (with A. Venkataraman) "The sine and cosine power series in Hindu mathematics", Journal of the Royal Asiatic SocietyEuler's identity (1,963 words) [view diff] exact match in snippet view article find links to article
e^{ix}=\cos x+i\sin x} where the inputs of the trigonometric functions sine and cosine are given in radians. In particular, when x = π, e i π = cos π +Linear encoder (1,612 words) [view diff] exact match in snippet view article find links to article
outputs. The industry standard analog output for linear encoders is sine and cosine quadrature signals. These are usually transmitted differentially soKaranapaddhati (1,362 words) [view diff] exact match in snippet view article find links to article
The following verse describes the infinite series expansions of the sine and cosine functions. cāpācca tattat phalato'pi tadvat cāpāhatāddvayādihatatCircular uniform distribution (952 words) [view diff] exact match in snippet view article find links to article
{S}}={\overline {R}}e^{i{\overline {\theta }}}} where the average sine and cosine are: C ¯ = 1 N ∑ n = 1 N cos ( θ n ) S ¯ = 1 N ∑ n = 1 N sin (Lemniscate elliptic functions (23,108 words) [view diff] exact match in snippet view article find links to article
lemn are used instead), are analogous to the trigonometric functions sine and cosine. While the trigonometric sine relates the arc length to the chord lengthHolonomic function (1,977 words) [view diff] exact match in snippet view article find links to article
rational functions the sine and cosine functions (but not tangent, cotangent, secant, or cosecant) the hyperbolic sine and cosine functions (but not hyperbolicKriyakramakari (1,126 words) [view diff] case mismatch in snippet view article find links to article
of the odd ones, [that] should be the arc. Here, the smaller of the Sine and Cosine is required to be considered as the desired [Sine]. Otherwise thereCurve fitting (2,139 words) [view diff] exact match in snippet view article find links to article
acceptable. Other types of curves, such as trigonometric functions (such as sine and cosine), may also be used, in certain cases. In spectroscopy, data may beUnit circle (1,009 words) [view diff] exact match in snippet view article find links to article
^{2}\theta +\sin ^{2}\theta =1.} The unit circle also demonstrates that sine and cosine are periodic functions, with the identities cos θ = cos ( 2 π kLaw of cosines (5,707 words) [view diff] exact match in snippet view article find links to article
of rectangle areas; Hellenistic trigonometry developed later, and sine and cosine per se first appeared centuries afterward in India. The cases of obtuseAll Alone (Dolo Coker album) (152 words) [view diff] case mismatch in snippet view article
compositions by Dolo Coker except as indicated "Reflections" - 5:54 "Sine and Cosine" - 2:15 "Just You" - 4:36 "Cabin in the Sky" (Vernon Duke, John LaExponential integral (2,995 words) [view diff] no match in snippet view article find links to article
retrieved 2011-08-09 Temme, N. M. (2010), "Exponential, Logarithmic, Sine, and Cosine Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, RonaldEccentric anomaly (1,148 words) [view diff] exact match in snippet view article find links to article
anomaly are related as follows. Using the formula for r above, the sine and cosine of E are found in terms of f : cos E = x a = a e + r cos f a =Hardy field (925 words) [view diff] exact match in snippet view article find links to article
multiplicative inverse or are zero. This means periodic functions such as the sine and cosine functions cannot exist in Hardy fields. This avoidance of periodicAir core gauge (438 words) [view diff] exact match in snippet view article find links to article
{x}{y}}\right)} Where x {\displaystyle x} and y {\displaystyle y} are the coils' sine and cosine currents respectively. The permanent magnet aligns itself with thatTimeline of mathematics (7,788 words) [view diff] exact match in snippet view article find links to article
numerical values. It defines the concepts of sine and cosine, and also contains the earliest tables of sine and cosine values (in 3.75-degree intervals fromIntegration using Euler's formula (1,135 words) [view diff] exact match in snippet view article find links to article
function and sine is odd. These two equations can be solved for the sine and cosine to give cos x = e i x + e − i x 2 and sin x = e i x − e − i x 2Logarithmic integral function (1,104 words) [view diff] no match in snippet view article find links to article
MR 0167642. LCCN 65-12253. Temme, N. M. (2010), "Exponential, Logarithmic, Sine, and Cosine Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, RonaldExponential function (5,755 words) [view diff] exact match in snippet view article find links to article
that, restricted to real arguments, the sine and cosine functions as defined above coincide with the sine and cosine functions as introduced in elementaryAlmost periodic function (2,838 words) [view diff] exact match in snippet view article find links to article
linear combination of sine and cosine waves that is of distance less than ε from f with respect to the uniform norm. The sine and cosine frequencies can beSquare and Compass, Worth Matravers (967 words) [view diff] case mismatch in snippet view article find links to article
Research Establishment scientists in Purbeck, who knew the inn as Sine and Cosine. The public house includes a small museum which displays fossils andCofunction (335 words) [view diff] exact match in snippet view article find links to article
Sine and cosine are each other's cofunctions.Lawrence Crawford (mathematician) (343 words) [view diff] case mismatch in snippet view article
They had three sons and two daughters. On the Use of the Hyperbolic Sine and Cosine (1895) The Tides (1897) The Trisection of a Given Angle (1898) EvaluationBounded function (985 words) [view diff] case mismatch in snippet view article find links to article
and Scientists, 5th Edition. CRC Press. ISBN 978-0-412-62150-5. "The Sine and Cosine Functions" (PDF). math.dartmouth.edu. Archived (PDF) from the originalGanita Kaumudi (1,160 words) [view diff] exact match in snippet view article find links to article
generalization here was crucial for finding the infinite series for sine and cosine. 28 rules and 19 examples. Geometry. 149 rules and 94 examples. IncludesAlternating series (1,693 words) [view diff] exact match in snippet view article find links to article
_{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}x^{n}\;=\;\ln(1+x).} The functions sine and cosine used in trigonometry and introduced in elementary algebra as the ratioQuaternions and spatial rotation (11,558 words) [view diff] exact match in snippet view article find links to article
intuitive and easy to understand and, due to the periodic nature of sine and cosine, rotation angles differing precisely by the natural period will beBhāskara I (1,346 words) [view diff] exact match in snippet view article find links to article
0 {\displaystyle x=0} ). Additionally, he gives relations between sine and cosine, as well as relations between the sine of an angle less than 90° andMathematical table (1,458 words) [view diff] exact match in snippet view article find links to article
culminating in the discovery of the power series expansions of the sine and cosine functions by Madhava of Sangamagrama (c.1350 – c.1425), and the tabulationSupertoroid (810 words) [view diff] exact match in snippet view article find links to article
similar to the trigonometric equations of the torus, except that the sine and cosine terms are raised to arbitrary powers. Namely, the generic point P(uKosambi–Karhunen–Loève theorem (10,711 words) [view diff] exact match in snippet view article find links to article
and the expansion basis consists of sinusoidal functions (that is, sine and cosine functions), the coefficients in the Karhunen–Loève theorem are randomLemniscate of Bernoulli (1,199 words) [view diff] exact match in snippet view article find links to article
The lemniscate sine and cosine relate the arc length of an arc of the lemniscate to the distance of one endpoint from the origin.Complex harmonic motion (963 words) [view diff] exact match in snippet view article find links to article
similarly to the simple pendulum because the motion is determined by sine and cosine functions as well. The image shows a marine clock with motor springsVincenzo Riccati (1,296 words) [view diff] exact match in snippet view article find links to article
modern publications, and formulated the addition laws of hyperbolic sine and cosine: C A = r , C B = C h . φ , B E = S h . φ , C D = C h . π ,Stumpff function (674 words) [view diff] exact match in snippet view article find links to article
{-x\ }}}~.\end{aligned}}} Circular orbits and elliptical orbits use sine and cosine relations, and hyperbolic orbits use the sinh and cosh relations. ParabolicFPS AP-120B (898 words) [view diff] exact match in snippet view article find links to article
the hardware was delivered with a pre-calculated lookup table of sine and cosine values. Sines and cosines for angles from 0 to π/2 radians were storedMurderous Maths (1,865 words) [view diff] exact match in snippet view article find links to article
right angled triangles, trig on a calculator; normal and inverse, sine and cosine formulas for non-right-angled triangles, triangulation, parallax anglesMarsaglia polar method (1,410 words) [view diff] exact match in snippet view article find links to article
{-2\ln(u_{1})}},} but that transform requires logarithm, square root, sine and cosine functions. On some processors, the cosine and sine of the same argumentAnalog computer (6,932 words) [view diff] exact match in snippet view article find links to article
that pin was a square block. Rectilinear-coordinate outputs (both sine and cosine, typically) came from two slotted plates, each slot fitting on thePythagorean theorem (12,566 words) [view diff] exact match in snippet view article find links to article
triangle with sides a, b and hypotenuse c, trigonometry determines the sine and cosine of the angle θ between side a and the hypotenuse as: sin θ = b cArea of a circle (5,880 words) [view diff] exact match in snippet view article find links to article
r^{2}} . This particular proof may appear to beg the question, if the sine and cosine functions involved in the trigonometric substitution are regarded asEuler angles (5,214 words) [view diff] exact match in snippet view article find links to article
=\arcsin \left(Y_{3}/{\sqrt {1-X_{3}^{2}}}\right).} Note that the inverse sine and cosine functions yield two possible values for the argument. In this geometricalEuler–Rodrigues formula (2,056 words) [view diff] exact match in snippet view article find links to article
φ is increased by a full rotation of 360 degrees, the arguments of sine and cosine only increase by 180 degrees. The resulting parameters are the oppositeDamping (2,749 words) [view diff] exact match in snippet view article find links to article
given sinusoidal waveform may be of intermediate phase, having both sine and cosine components. The term "damped sine wave" describes all such damped waveformsWeierstrass factorization theorem (1,872 words) [view diff] exact match in snippet view article find links to article
}E_{p_{n}}\!\!\left({\frac {z}{a_{n}}}\right).} The trigonometric functions sine and cosine have the factorizations sin π z = π z ∏ n ≠ 0 ( 1 − z n ) e z / nCollatz conjecture (7,090 words) [view diff] exact match in snippet view article find links to article
to the complex plane. They used Chamberland's function for complex sine and cosine and added the extra term 1 π ( 1 2 − cos ( π z ) ) sin ( π z )Prosthaphaeresis (1,972 words) [view diff] exact match in snippet view article find links to article
prosthaphaeresis with values for every second, or 3600th of a degree. Inverse sine and cosine functions are particularly troublesome, because they become steep nearHelmholtz equation (2,973 words) [view diff] exact match in snippet view article find links to article
equation in time. The solution in time will be a linear combination of sine and cosine functions, whose exact form is determined by initial conditions, whileIntegration by substitution (3,322 words) [view diff] exact match in snippet view article find links to article
be integrated using substitution by expressing it in terms of the sine and cosine: tan x = sin x cos x {\displaystyle \tan x={\tfrac {\sin x}{\cosRC oscillator (1,957 words) [view diff] exact match in snippet view article find links to article
It is possible to stabilize a quadrature oscillator by squaring the sine and cosine outputs, adding them together, (Pythagorean trigonometric identity)Lissajous-toric knot (933 words) [view diff] exact match in snippet view article find links to article
Lissajous-toric knot onto the x-y-plane is a Lissajous curve. Replacing the sine and cosine functions in the parametrization by a triangle wave transforms a Lissajous-toricSpring (device) (3,353 words) [view diff] exact match in snippet view article
{d^{2}x}{dt^{2}}}+{\frac {k}{m}}x=0,\,} the solution of which is the sum of a sine and cosine: x ( t ) = A sin ( t k m ) + B cos ( t k m ) . {\displaystyle x(t)=A\sinCauchy sequence (3,219 words) [view diff] exact match in snippet view article find links to article
Golden ratio, which is irrational. The values of the exponential, sine and cosine functions, exp(x), sin(x), cos(x), are known to be irrational for anyEvoked potential (5,213 words) [view diff] exact match in snippet view article find links to article
flickering (sinusoidally modulated) light. Rather than integrating the sine and cosine products, Regan fed the signals to a two-pen recorder via lowpass filtersRotary encoder (2,819 words) [view diff] exact match in snippet view article find links to article
sine encoders, which output sinusoidal waveforms in quadrature (i.e., sine and cosine), thus combining the characteristics of an encoder and a resolver.Lookup table (3,070 words) [view diff] exact match in snippet view article find links to article
floating-point divide bug. Functions of a single variable (such as sine and cosine) may be implemented by a simple array. Functions involving two or moreList of common coordinate transformations (2,162 words) [view diff] exact match in snippet view article find links to article
In this case in all formulas below all arguments in θ should have sine and cosine exchanged, and as derivative also a plus and minus exchanged. All divisionsRangekeeper (4,719 words) [view diff] exact match in snippet view article find links to article
computer multipliers were based on the geometry of similar triangles. Sine and cosine generation (polar-to-rectangular coordinate conversion) These mechanismsHistory of mathematics (16,142 words) [view diff] exact match in snippet view article find links to article
the Madhava-Newton power series to determine sine and cosine and the Taylor approximation for sine and cosine functions. In the 16th century, JyesthadevaPeskin–Takeuchi parameter (1,801 words) [view diff] exact match in snippet view article find links to article
}(0)-s_{w}^{2}\Pi _{\gamma \gamma }^{\prime }(0)\right]} where sw and cw are the sine and cosine of the weak mixing angle, respectively. The definitions are carefullyBox–Muller transform (2,089 words) [view diff] exact match in snippet view article find links to article
R2, which is likewise uniformly distributed. The definitions of the sine and cosine are then applied to the basic form of the Box–Muller transform to avoidInterval arithmetic (8,154 words) [view diff] exact match in snippet view article find links to article
where the monotonicity of the function changes direction. For the sine and cosine functions, the critical points are at ( 1 2 + n ) π {\displaystyleAryabhata (4,786 words) [view diff] no match in snippet view article find links to article
90°, to an accuracy of 4 decimal places. In fact, the modern terms "sine" and "cosine" are mistranscriptions of the words jya and kojya as introduced byLine (geometry) (4,301 words) [view diff] exact match in snippet view article
be proven geometrically by applying right triangle definitions of sine and cosine to the right triangle that has a point of the line and the origin asEquation (4,237 words) [view diff] exact match in snippet view article find links to article
manipulating or solving trigonometric equations. Two of many that involve the sine and cosine functions are: sin 2 ( θ ) + cos 2 ( θ ) = 1 {\displaystyle \sinLogarithm (11,600 words) [view diff] exact match in snippet view article find links to article
r={\sqrt {x^{2}+y^{2}}}.} Using the geometrical interpretation of sine and cosine and their periodicity in 2π, any complex number z may be denoted asModulation (3,050 words) [view diff] exact match in snippet view article find links to article
component. Carry out the modulation, for example by multiplying the sine and cosine waveform with the I and Q signals, resulting in the equivalent lowTheorem (4,394 words) [view diff] exact match in snippet view article find links to article
{\displaystyle \tan(\alpha +\beta )} from the addition formulas of sine and cosine. Petkovsek et al. 1996. Wentworth, G.; Smith, D.E. (1913). Plane GeometryAN/FSQ-7 Combat Direction Central (2,372 words) [view diff] exact match in snippet view article find links to article
register which was incremented 32 times a second.: 27 Trigonometric sine and cosine functions used 1.4 degree precision (256 values) via look-up tablesWheel speed sensor (2,793 words) [view diff] exact match in snippet view article find links to article
magneto-resistive sensor scans a target wheel with 256 teeth, generating sine and cosine signals. Arctangent interpolation is used to generate rectangular pulsesTrigonometric integral (3,230 words) [view diff] no match in snippet view article find links to article
Differential Equations. Temme, N.M. (2010), "Exponential, Logarithmic, Sine, and Cosine Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, RonaldMadhava of Sangamagrama (3,711 words) [view diff] exact match in snippet view article find links to article
unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine", Historia Scientiarum, 42: 49–65. Biography on MacTutorSeasonality (2,502 words) [view diff] exact match in snippet view article find links to article
sinusoidal model, Fourier terms added into regression models utilize sine and cosine terms in order to simulate seasonality. However, the seasonality ofTrend periodic nonstationary processes (966 words) [view diff] exact match in snippet view article find links to article
periodic stationary processes is by using a regression model with a sine and cosine function, such as: x ( t ) = β 0 + β 1 t + β 2 c o s ( 2 π t / T )Eclipse cycle (10,441 words) [view diff] exact match in snippet view article find links to article
and distance of the Moon and of the Sun using sine and cosine series. The arguments of the sine and cosine functions depend on only four values, the DelaunayGlossary of electrical and electronics engineering (19,294 words) [view diff] exact match in snippet view article find links to article
mathematical technique for representing a sampled signal as a sum of sine and cosine waves of different frequencies. discrete-time signal A signal representedVibrations of a circular membrane (2,508 words) [view diff] exact match in snippet view article find links to article
convenience. Then, T ( t ) {\displaystyle T(t)} is a linear combination of sine and cosine functions, T ( t ) = A cos c λ t + B sin c λ t . {\displaystyleBicentric quadrilateral (3,735 words) [view diff] exact match in snippet view article find links to article
{cd}{ab}}}=\cot {\frac {D}{2}}.\end{aligned}}} Using the same notations, for the sine and cosine functions the following formulas holds: sin A 2 = b c a d + b c =Circle (6,119 words) [view diff] exact match in snippet view article find links to article
can be written in parametric form using the trigonometric functions sine and cosine as x = a + r cos t , y = b + r sin t , {\displaystyle {\begin{aligned}x&=a+r\Single-sideband modulation (4,797 words) [view diff] exact match in snippet view article find links to article
the middle of the voiceband, but implemented by a quadrature pair of sine and cosine modulators at that frequency (e.g. 2 kHz). This complex signal or pairRadian (5,387 words) [view diff] exact match in snippet view article find links to article
angles are involved, mathematically important relationships between the sine and cosine functions and the exponential function (see, for example, Euler's formula)Underwater acoustic communication (1,927 words) [view diff] exact match in snippet view article find links to article
signal is impressed into the magnetic field x,y area by varying the sine and cosine inputs at a precise time. It is widely used for wireless LANs , RFIDSquigonometry (1,107 words) [view diff] exact match in snippet view article find links to article
corresponds to s q p ( t ) {\displaystyle sq_{p}(t)} . The definition of sine and cosine through integrals can be extended to define the squigonometric functionsFunction of a real variable (3,560 words) [view diff] exact match in snippet view article find links to article
polynomial functions, including constant functions and linear functions Sine and cosine functions Exponential function Some functions are defined everywhereCis (mathematics) (2,340 words) [view diff] exact match in snippet view article
platform the fused operation is about twice as fast as calling the sine and cosine functions individually. d d z cis z = i cis z = i e i z {\displaystyleBounded type (mathematics) (1,666 words) [view diff] exact match in snippet view article
with Q ( z ) = exp ( | a | i z ) {\displaystyle Q(z)=\exp(|a|iz)} . Sine and cosine are of bounded type in the UHP. Indeed, sin ( z ) = P ( z ) / Q (Binomial theorem (6,250 words) [view diff] exact match in snippet view article find links to article
with de Moivre's formula to yield multiple-angle formulas for the sine and cosine. According to De Moivre's formula, cos ( n x ) + i sin ( n x )Weierstrass elliptic function (4,401 words) [view diff] exact match in snippet view article find links to article
\quad t\mapsto (\sin t,\cos t).} Because of the periodicity of the sine and cosine R / 2 π Z {\displaystyle \mathbb {R} /2\pi \mathbb {Z} } is chosenSquare root of 2 (5,701 words) [view diff] exact match in snippet view article find links to article
= 1/√2, along with the infinite product representations for the sine and cosine, leads to products such as 1 2 = ∏ k = 0 ∞ ( 1 − 1 ( 4 k + 2 ) 2 )Taylor series (8,253 words) [view diff] exact match in snippet view article find links to article
polynomials, exponential function ex, and the trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are notIndian people (7,621 words) [view diff] exact match in snippet view article find links to article
equations. Chakravala method, sign convention, madhava series, and the sine and cosine in trigonometric functions can be traced to the jyā and koti-jyā. NotablePi (17,481 words) [view diff] exact match in snippet view article find links to article
trigonometric functions have periods that are multiples of π; for example, sine and cosine have period 2π, so for any angle θ and any integer k, sin θ = sinLists of integrals (5,611 words) [view diff] exact match in snippet view article find links to article
step function. If we also use the fact that the absolute values of sine and cosine are periodic with period π, then we get: ∫ | sin a x | d x = 2 aSynchronous coefficient of drag alteration (2,152 words) [view diff] exact match in snippet view article find links to article
using trigonometric identities results in a sum of constant terms, sine and cosine, at angular frequency 2 ω {\displaystyle 2\omega } . The next calculationsIntegration by reduction formulae (2,235 words) [view diff] exact match in snippet view article find links to article
following integrals contain: Factors of sine Factors of cosine Factors of sine and cosine products and quotients Products/quotients of exponential factors andBioche's rules (1,160 words) [view diff] exact match in snippet view article find links to article
{\displaystyle \sin t,\tan t,\cos(2t),\tan(t/2)} ), in the case of hyperbolic sine and cosine, a good change of variable is u = cosh ( t ) {\displaystyle u=\cosh(t)}Lode coordinates (1,685 words) [view diff] exact match in snippet view article find links to article
\theta _{s}} can be shown by applying a trigonometric identity relating sine and cosine by a shift cos ( 3 θ c ) = sin ( 3 θ s ) {\displaystyle \cos(3\thetaPauli matrices (7,437 words) [view diff] exact match in snippet view article find links to article
{\sigma }}\,.} Matrix exponentiating, and using the Taylor series for sine and cosine, e i a ( n ^ ⋅ σ → ) = ∑ k = 0 ∞ i k [ a ( n ^ ⋅ σ → ) ] k k ! = ∑Integration by parts (6,875 words) [view diff] exact match in snippet view article find links to article
{\displaystyle v^{(n)}} are readily available (e.g., plain exponentials or sine and cosine, as in Laplace or Fourier transforms), and when the nth derivativeL'Hôpital's rule (7,121 words) [view diff] exact match in snippet view article find links to article
many common functions have continuous derivatives (e.g. polynomials, sine and cosine, exponential functions), so this special case covers most applicationsPolar coordinate system (6,687 words) [view diff] exact match in snippet view article find links to article
Cartesian coordinates x and y by using the trigonometric functions sine and cosine: x = r cos φ , y = r sin φ . {\displaystyle {\begin{aligned}x&=r\cosScrew theory (4,395 words) [view diff] exact match in snippet view article find links to article
ẑ = (φ, d) define a dual angle, then the infinite series definitions of sine and cosine yield the relations sin z ^ = ( sin φ , d cos φ ) , cos z ^Casus irreducibilis (3,373 words) [view diff] exact match in snippet view article find links to article
roots and trisectability can also be extended to some cases where the sine and cosine of the given angle is irrational. Consider as an example the case whereGupta Empire (8,556 words) [view diff] exact match in snippet view article find links to article
Varāhamihira wrote the Pancha Siddhanta developing various formulas relating sine and cosine functions. Yativṛṣabha made contributions on units of measurement.Mathematics (15,696 words) [view diff] exact match in snippet view article find links to article
Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series. During the Golden Age of IslamWave (7,836 words) [view diff] exact match in snippet view article find links to article
combination of two sine waves with phases of zero and a quarter cycle, the sine and cosine components, respectively. A plane wave is a kind of wave whose valueProof that π is irrational (5,785 words) [view diff] exact match in snippet view article find links to article
have F ″ + F = f . {\displaystyle F''+F=f.} The derivatives of the sine and cosine function are given by sin' = cos and cos' = −sin. Hence the productHankel transform (4,204 words) [view diff] exact match in snippet view article find links to article
1145/317275.317284. Knockaert, Luc (2000). "Fast Hankel transform by fast sine and cosine transforms: the Mellin connection". IEEE Trans. Signal Process. 48Function (mathematics) (11,317 words) [view diff] exact match in snippet view article
Graph of two trigonometric functions: sine and cosine.Formative assessment (7,558 words) [view diff] exact match in snippet view article find links to article
looking to each other to gain knowledge. "You seem to be confusing sine and cosine. Talk to Katie about the differences with the two." "Compare your workList of incomplete or incorrect mathematical proofs (4,299 words) [view diff] exact match in snippet view article find links to article
definition of pointwise convergence. For example, a Fourier series of sine and cosine functions, all continuous, may converge pointwise to a discontinuousAssociated Legendre polynomials (5,475 words) [view diff] exact match in snippet view article find links to article
there are 2ℓ + 1 functions for the various values of m and choices of sine and cosine. They are all orthogonal in both ℓ and m when integrated over the surfaceSeries (mathematics) (9,694 words) [view diff] exact match in snippet view article
expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jacob Bernoulli (1702) and his brotherList of Egyptian inventions and discoveries (17,995 words) [view diff] exact match in snippet view article find links to article
formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine. Ptolemy furtherOort constants (4,693 words) [view diff] exact match in snippet view article find links to article
}_{R_{0}}d\cdot \cos ^{2}\left(l\right)-\Omega d\\\end{aligned}}} Using the sine and cosine half angle formulae, these velocities may be rewritten as: V obs, rNormal distribution (22,467 words) [view diff] exact match in snippet view article find links to article
of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, U and V are drawn from the uniform (−1,1)Timeline of scientific discoveries (10,369 words) [view diff] exact match in snippet view article find links to article
to 5th centuries: The modern fundamental trigonometric functions, sine and cosine, are described in the Siddhantas of India. This formulation of trigonometrySpinodal decomposition (6,641 words) [view diff] exact match in snippet view article find links to article
mathematically represented by sines and cosines. Due to the properties of sine and cosine, it is possible to recover the amount of each wave in the sum by anSpherical harmonics (12,420 words) [view diff] exact match in snippet view article find links to article
\theta )\ \cos(m\varphi )&{\text{if }}m>0\,.\end{cases}}} The same sine and cosine factors can be also seen in the following subsection that deals withHyperbola (13,551 words) [view diff] exact match in snippet view article find links to article
{\displaystyle y=1/x} is a rectangular hyperbola. Using the hyperbolic sine and cosine functions cosh , sinh {\displaystyle \cosh ,\sinh } , a parametricList of British innovations and discoveries (13,433 words) [view diff] exact match in snippet view article find links to article
multiplication as well as the abbreviations "sin" and "cos" for the sine and cosine functions – William Oughtred Discovery of the "White Spot" on SaturnHistory of Grandi's series (6,535 words) [view diff] case mismatch in snippet view article find links to article
authors list (link) Earnshaw, Samuel (1844-12-09). "On the Values of the Sine and Cosine of an Infinite Angle". Transactions of the Cambridge PhilosophicalEdward Sang (5,913 words) [view diff] exact match in snippet view article find links to article
349–352 link Edward Sang, On a new investigation of the series for the sine and cosine of an arc, Proceedings of the Royal Society of Edinburgh, volume 9Rule of marteloio (8,275 words) [view diff] exact match in snippet view article find links to article
although trigonometry was only in its infancy in Christian Europe, sine and cosine tables were already known in Arab mathematics. The Kingdom of MajorcaSingular spectrum analysis (6,725 words) [view diff] exact match in snippet view article find links to article
Fourier Analysis In contrast with Fourier analysis with fixed basis of sine and cosine functions, SSA uses an adaptive basis generated by the time seriesList of English inventions and discoveries (16,728 words) [view diff] exact match in snippet view article find links to article
symbol for multiplication and the abbreviations "sin" and "cos" for the sine and cosine functions devised by William Oughtred (1574–1660) in Clavis MathematicaeHistory of mathematical notation (11,512 words) [view diff] exact match in snippet view article find links to article
his proportionality sign (∷) and abbreviations sin and cos for the sine and cosine functions. Albert Girard also used the abbreviations 'sin', 'cos' andX86 instruction listings (12,793 words) [view diff] exact match in snippet view article find links to article
compliant floating-point partial remainder. FPREM1 D9 F5 Floating-point sine and cosine. Computes two values S = sin ( k ∗ s t ( 0 ) ) {\displaystyle S=\sinMarcum Q-function (7,425 words) [view diff] exact match in snippet view article find links to article
express the Bessel functions with half-integer as sum of hyperbolic sine and cosine functions: I n + 1 2 ( z ) = 2 z π [ g n ( z ) sinh ( z ) + g − nBasil Hiley (11,854 words) [view diff] exact match in snippet view article find links to article
operator equations are analogous to the two equations that involve the sine and cosine bracket, and that the quantum phase equation has apparently not beenFokas method (5,052 words) [view diff] exact match in snippet view article find links to article
{\displaystyle \lambda } -plane, wheres the analogous equations for the sine and cosine transforms are valid only for λ {\displaystyle \lambda } real. TheProduct operator formalism (5,744 words) [view diff] exact match in snippet view article find links to article
the final equality follows from recognising the Taylor series for sine and cosine. Now suppose that the density operator at time zero is ρ ( 0 ) = A