What is Sine: Definition and 520 Discussions

In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle (the hypotenuse). For an angle



x


{\displaystyle x}
, the sine function is denoted simply as



sin
⁡
x


{\displaystyle \sin x}
.More generally, the definition of sine (and other trigonometric functions) can be extended to any real value in terms of the length of a certain line segment in a unit circle. More modern definitions express the sine as an infinite series, or as the solution of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.
The sine function is commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year.
The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic, and then from Arabic to Latin. The word "sine" (Latin "sinus") comes from a Latin mistranslation by Robert of Chester of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.

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  1. A

    Voltage, Current, and Time Graph with Sine Wave

    Homework Statement Hello all! Basically, my professor used this graph in our lecture, but I do not really understand how to read it. The graph has three variables, so I'm guessing it is Voltage vs. Current as time increases? One question he asked was "The figure shows the voltage and...
  2. V

    Conversion to phasor from cos and sine

    I am given the problem: 100Sin(ωt+30) +20cos(ωt) and the solution is 78.7<38.9degrees How do I convert to phasor form? I know for this calculation I need the following relationship: √2Esin(ωt-∅)=Ee^(-j∅) At first I tried making both sines and then splitting up 100Sin(ωt+30) and...
  3. P

    Fitting a sine curve that isn't a perfect sine curve?

    For my 3rd year project, I have a set of data which is the variation within an interference pattern. The outcome has clear sine curve properties (Alternating peaks and troughs of intensity) but the positions aren't regular (i.e. the peak-to-peak distance between peak1 and peak2 is larger than...
  4. O

    Converting Modifed Sine Wave to Sine Wave

    Good evening, I bought my self my first Inverter rated at 500W which has a peak rate of 1000W (1kW) maximum. Since some devices such as my main computer for one takes 600W, It does not like the modified sine wave. I was wondering if there is a way to somehow 'convert' the modified sine wave...
  5. A

    Sine function as an infinite sequence

    I have a vague understanding of how to derive the sine function from a Maclaurin Sequence however this isn't helping me figure out why: (1 - \frac{x^{2}}{4π^{2}}) (1 - \frac{x^{2}}{9π^{2}}) (1 - \frac{x^{2}}{16π^{2}})... = \frac{π^{2}}{x(x+π)}\frac{sin x - sin π}{x - π} Any help would be...
  6. 1

    Turn Envelope Wave into one Sine Wave

    Hello everyone, The sum of two sine waves that have close frequencies can produce an Envelope Wave: http://en.wikipedia.org/wiki/Envelope_(waves) What formula can I use to turn it into one wave (i.e. one frequency and amplitude)? Example pic attached: The blue "Envelope Wave" is...
  7. P

    Need help determining whether to use cosine or sine

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  8. Petrus

    MHB Help with Solving Limit of Cosine and Sine

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  9. L

    Help with computing/understanding Fourier Sine Expansion.

    1. Find the Fourier sine expansion of \phi(x)=1. This was posted in Calculus and Beyond thread, but I realized that this thread may be more appropriate. 2. The attempt at a solution. I start with \phi(x)=A_1sin(\pi x)+A_2sin(2\pi x)+\cdots+A_nsin(n\pi x), and then add multiply by A_msin(m\pi x)...
  10. L

    Help with Fourier Sine Expansion.

    1. Find the Fourier sine expansion of \phi(x)=1 .2. Homework Equations . In the lecture the professor worked his stuff from scratch so I was trying to do it like his example.3. The Attempt at a Solution . I start with \phi(x)=A_1sin(\pi x)+A_2sin(2\pi x)+\cdots+A_nsin(n\pi x), and then add...
  11. G

    Sine Wave between two Functions

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  12. A

    Understanding the Fourier Sine Transform: Valid Inputs and Applications

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  13. R

    Proof the identities of the sine and cosine sum of angles

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  14. F

    Addition of Sine & Cosine

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  15. A

    What is the connection between sine and cosine and geometry?

    Ordinarily in mathematics, when you want to define a function, it is without reference to geometry. For instance the mapping f:ℝ→ℝ x→x2 And though I don't know much about mathematics I assume you somehow proof that the function is well defined for all numbers, check if the derivative exists and...
  16. R

    Determining the possible time interval between two sine waves

    ok I just got stuck half way into a problem, I would like it if someone explained it! Ok the question says, two identical sinusodial waves with a wavelength of 3.0 m and traveling in the same direction with a v of 2m/s. Starting from the same point, just the second waves starts later. and the...
  17. jbrussell93

    Optics: Complex exponentials for sine

    Homework Statement Imagine that we have two waves of the same amplitude, speed and frequency over-lapping in some region of space such that the resultant disturbance is \psi(y,t) = Acos(ky+\omega t) + Acos(ky-\omega t +\pi) Using complex exponentials show that \psi(y,t) = -2Asin(ky)...
  18. P

    What is the meaning of the y-axis on a sine wave graph?

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  19. R

    How do you find the phase difference when given two sine equations and a X and t

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  20. S

    Minimize a certain function involving sine and cosine

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  21. F

    First nonzero terms of Fourier sine series

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  22. D

    What causes the oscillations in an RLC circuit?

    So I'm trying to get my head around RLC circuits and what is happening say we have a 10 ohm resistor, 1000uf capacitor and a 10H inductor in a circuit powered by a 10 volt source. Is the resulting oscillation of the sine wave frequency the result of the inductor resisting the flow of...
  23. D

    PDE involving Fourier Sine Series

    Homework Statement Solve: PDE: 9Uxx=Utt BC: U(0, t) = U(∏, t) = 0 IC: U(x, 0) = sin4x + 7sin5x Ut = x 0 < x < ∏/2 = ∏ - x ∏/2 < x < ∏ Homework Equations Fourier Sine Series and Cosine Series Equations The Attempt at a Solution...
  24. D

    Fourier Sine Series of f(x)=x: Solution & Explanation

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  25. E

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  26. P

    Sine Ratio Test: Show Convergence w/o L'Hopital

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  27. T

    Why do sine waves define pure frequency?

    Like the title says, I'm curious why sine waves are often referred to as "natural" or "pure" oscillations. Why not some other oscillating function? As an example of the type of idea I'm referring to, the wikipedia "Sine Wave" article says: To me, the argument that it sounds pure because...
  28. K

    Fourier Series- half range sine series

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  29. B

    Laplace Sine Transform: Re[s]>0

    L[sin(at)]=\frac{a}{s^{2}+a^{2}}, Re[s]>0 L[e^{kt}]=\frac{1}{s-k}, s>k L[e^{-kt}]=\frac{1}{s+k}, s<-k L[sin(at)]=\frac{1}{2i}L[e^{iat}-e^{-iat}] =\frac{1}{2i}L[e^{iat}]-L[e^{-iat}] Using the above relations =\frac{1}{2i}[\frac{1}{s-ia}-\frac{1}{s+ia}], s>ia, s<-ia The problem is...
  30. dexterdev

    Why are there more zeros in a sine wave than 1s and -1s?

    Hello guys, From a layman's viewpoint when we count occurrences of different amplitude values, then frequency of zeros must be maximum is not it? but it is as below: why so? zeoros must be maximum, 1s and -1s are lesser than 0 isn't it?
  31. T

    Understanding the Proof of Cosine and Sine Rules: Am I on the Right Track?

    At the moment I a trying to really get to grips with these rules proof wise. I would like to know if I am on the right track in learning the proof correctly. In my books it goes through the proof in breaking up an acute and obtuse triangle and the solving via trigonometry/Pythagoras. I...
  32. E

    How i can plot anti periodic sine wave in matlab ?

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  33. A

    DFT of Sine Function: Why 2 Values at K=K0 & -K0?

    why does the DFT of sin(2∏nKo/N) give two value at k=K0 and k=-K0 though sin has only one frequency component and the DFT is complex , what does it mean? frequency can't have any phase the why it is complex? thanks
  34. Y

    Sine Wave measurements vs equations

    I have the equation E(t)=7*sin(11000t+∏/3) and I measured the following: E(.22ms) = .422V Frequency = 1.751kHz Period = 572.0 μs Peak = 7V Peak-Peak = 14V E(rms) = 4.8V E(average) = 105mV I've calculated the following to compare E(.22ms) = .422V Frequecy = 1.75 kHz Period...
  35. E

    Learn About Sine Wave Separation and Sound Processing

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  36. D

    MHB Special cases for sine and cosine sum

    State the special cases of the above two formulas for $n = 0, 1,$ and $2$. These should be familiar formulas. I don't see what is so special and familiar about when n = 2 or for cosine n = 1.When $n = 0$, we have $$ \sum\limits_{k = 0}^0\cos k\theta =...
  37. S

    Question using the Sine Rule and Bearings

    Homework Statement 3 towns are located at D,F & E; the vertices of an scalene triangle. The bearing of F from D is 48 degrees. D to E=640m and E to F=430m. Point G is located at the end of a south line at F. a)i)Find angle DFG ii) Find angle GFE iii) Find angle DFE b) Calculate angle...
  38. H

    Sine Function with Alternating Peaks and Wavelengths

    Hi All, I'm trying to find a sine function that models the cyclic and seasonal nature of retail sales across the years. I've tried several combinations including some basic attempts at complex plots of sin(x), but am now starting to wonder if there is even a simple solution to the problem...
  39. S

    Is Sqrt(3) Factor in 3-Phase Power Dependent on Sine Waves?

    The line current in a 3-phase delta loaded circuit is supposedly equal to sqrt(3) times the phase current. That's dependent on all the signals being sine waves though right? Because the math that gets you to that point is dependent on sine and cosine identities that don't necessarily hold with...
  40. dexterdev

    Why sine wave is the basic building block signal to make other signals?

    Why sine wave is the basic building block signal to make other signals? Not any other wave. -Devanand T
  41. 1

    Physics textbook, replace sine with its definition (?)

    physics textbook, replace sine with its definition (?) What on Earth do they mean? "That will introduce either sin(theta) or cos(theta). Reduce the resulting two variables, x and theta, to one, x, by replacing the trigonometric function with an expression (its definition) involving x and y."...
  42. A

    Derivation of formula for general formula of sine equations

    My book gives the following formulas for the general solution of sine equations (if sin(x)=a) x=2n\pi + sin^-^1(a) and x=(2n+1)\pi - sin^-^1(a) Alternatively x=n\pi + (-1)^nsin^-^1(a) But it doesn't explain how it got them. I can easily see how they got the first two but I have no idea how...
  43. F

    What do the terms LINE and SINE mean?

    in this picture, there is a legend with the terms LINE and SINE in it http://i.imgur.com/IGJfm.png what do the terms LINE and SINE mean? I mean what do they stand for and what is their purpose in DNA. I have tried to look them up but it keeps going to math.
  44. S

    How to construct a Modified sine wave

    Homework Statement Is it possible to generate a simple modified wave using simple circuit design? I know there is plenty of inverter available in the market, but I wish I can construct 1 by myself. How can I get started? Homework Equations The Attempt at a Solution
  45. D

    Why sine is used for cross product and cosine for dot product?

    While we calculate cross product of two vectors let A and B we write ABsinθ. And while we calculate dot product of them we write ABcosθ. Why particularly we use sinθ for cross product and cosθ for dot product.Is there any physical reason why we choose sine for cross product and cosine for...
  46. T

    Definite integral evaluation (Inverse Fourier Sine Transform)

    Homework Statement I'm working on a long problem and have come to the final step. The answer seems so simple, but I can't quite get to it. I need to evaluate this integral: \int_0^{\infty}\ \left(e^{-k^2 t}/k\right)\sin(kx)\ dk Homework Equations Mathematica gives the result as...
  47. E

    Using rotating vector to draw sine waves

    We have met the idea that a radius of a circle rotating ANTI-CLOCKWISE can be used to draw a sine wave... I get that... it is a great idea but...why does it have to be rotating anti-clockwise. That seems so un natural to me. We were told it is a convention. Does that mean it is something...
  48. C

    Is the Sine Function Linear in Small Domains?

    Sine function linear?? I have a problem concerning trigonometry and calculus but I just need to know my question's answer to solve it. I would like to know: can a sine function be construed as a linear function in a very small domain i.e increments of 0.0001?? Thank you so much in advance and...
  49. P

    How is sine of 90 degree possible

    I don't know how can a triangle be formed with two angles each 90 degree. If it is not possible, then how do we calculate sin 90 degree = 1 ?
  50. L

    Solving Sine Waves Problem: Find Time for π Radians

    A sine wave with a frequency of 10Hz and an amplitude of 5 units phases from zero to 2π radians in a complete cycle. How long does it take for the phase to reach a value of π radians? y(t) = A . sin (2.π.f.t + φ) 0.1 = 5 x sin (2x∏x10x time + 3.14) time = 0.038 seconds But the answer...
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