Sine Definition and 521 Threads

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. B

    Group velocity of two superimposed sine waves

    Hi all, I understand the concept of group velocity when applied to superimposed sine waves of the same amplitude, and even when applied to wave packets (in which case you get the well-known expression ∂ω/∂k). My question is what happens when you add two sine waves of different amplitudes? So...
  2. T

    Orthogonality of sine and cosine question

    Hello, I'm trying to solve Fourier Series, but I have a question. I know that cos(nx) is even and sin(nx) is odd. But what does this mean when I take the integral or sum of cos(nx) or sin(nx)? Do they have a value or do they just keep their form?
  3. DreamWeaver

    MHB The Multiple Sine function

    Please excuse the usual rambling preamble (or should that be pre-ramble?), but last year, when idly searching on-line, I happened to chance upon a truly great, great paper by Shin-ya Koyama and Nobushige Kurokawa, concerning the "Multiple Sine function", \mathscr{S}_n(x). The (free) paper in...
  4. N

    Alternative deduction of sum of sine and cosine

    Hi! Many students know that A\sin(x) + B\cos(x) =\sqrt{A^2+B^2} \sin{(x+\arctan \frac{B}{A})}. I have seen just one deduction of that relation, showed by set up a system of two equations, solving for amplitude and phase shift. Is it possible to deduce the relation in a vectorial way, or in...
  5. E

    MHB Determine the unknown angle (using the sine law)

    So using the sine law, I found that Θ = 68 degrees. And the I found that the other possibility of Θ would be 112 degrees (180 - 68 = 112). However, the textbook says that the answers are Θ = 68 and Θ = 23. What did I do wrong? Thanks.
  6. Q

    Question about rms value of a sine wave

    Sorry if this sounds like a dumb question, but why is the effective value of a sine wave 0.707, as opposed to 0.637 which is the value generated by finding the definite integral over the domain [0,∏] divided length of the domain?
  7. B

    Trig- Sine graph translations question help (C2 A level)

    Please see question 9ii: Paper: http://www.edexcel.com/migrationdocuments/QP%20GCE%20Curriculum%202000/January%202012%20-%20QP/6664_01_que_20120307.pdf Mark scheme: http://www.edexcel.com/migrationdocuments/QP%20GCE%20Curriculum%202000/January%202012%20-%20MS/6664_01_msc_20120123.pdf This is my...
  8. O

    LM324: How to Create Sine Wave Output

    Hello i would like to know, how does LM324 create a sine wave after the output of the op amp?? thanks
  9. Yae Miteo

    Integration by parts with e and sine

    Homework Statement Evaluate the integral. Homework Equations \int e^{2x} sin(3x) dx The Attempt at a Solution I began by using integration by parts. u = sin(3x) v = \frac {e^{2x}} {2} du = 3 cos(3x) dv = e^{2x} dx but I get stuck after that because the...
  10. S

    MATLAB Measuring the 'purity' of a sine wave using FFT (Matlab)

    I have a range of sine waves I have obtained in an experiment. I want to put a measure on the purity of these sine waves - how well the reproduce a theoretical sine wave. Is there anyway I can analyse the FFT of the sine waves in Matlab and put a measure on the purity of the sine wave...
  11. Newtons Apple

    Understanding the Relationship Between Sine Waves and Sine Angles?

    Hello everyone. Can someone explain the relationship between the idea of a sine wave, and the idea of a sine angle? I'm getting into trig, and I hear both terms of sin tossed around, but they seem to be completely unrelated. What does the angle of the triangle have to do with a wave? Same...
  12. M

    Combining Sine Waves: Adding 2 sin (314.2t) and 2 sin(314.2t - 120°)

    I need to add together the following sine waves and express the answer in the same form: Va = 2 sin (314.2t) + Vb = 2 sin(314.2t - 120°) Any help would be greatly appreciated.
  13. R

    Is there any symmetry I can use to find this Fourier sine series?

    Homework Statement I am going over a practice exam, and I need to find the FSS of f(x)=x(\pi^2-x^2) Homework Equations f(x) \sim \sum^\infty_{n=1}a_n sin\left(\frac{n \pi x}{L}\right) a_n=\frac{2}{L}\int^L_0 f(x)sin\left(\frac{n\pi x}{L}\right)dx The Attempt at a Solution I think I...
  14. A

    Sine wave power amplification question

    Hello all, I'm new to electrical engineering, and I'm working on a project that requires a bit more than I'm familiar with. What I'm asking for help on boils down to this: I have a circuit that outputs a low power sine wave (frequency 25-40kHz) with an amplitude +/- 5V at a few watts. I want to...
  15. P

    Hyperbolic Sine - Exponent transition

    Hey I didn't understand the transition below, I'd be glad for some help thanks
  16. C

    Finding Volume Using the Sine Function and Disc Method

    Homework Statement Find the volume generated by revolving one arch of the curve y = 5 sin(x) about the x-axis. The attempt at a solution So I figured this would create a disc so I would have to use that the volume is ∏r^2 where r=sinx, r^2= (sin(x))^2 and that the way I should...
  17. T

    Java Sine wave equation into Java Code

    I have been trying to implement this Wave equation into java: A = amplitude of wave L = wave length w = spatial angular frequency s = speed wt = temporal angular frequency d = direction FI = initiatory phase Y（x，y，t）=A*cos（w *（x，y）+ wt*t + FI; I...
  18. N

    Use LabView to simulate 2 Sine Waves and then Filter Waves above 50Hz

    Homework Statement Create a VI that will: simulate two sine waves of different frequencies (run it for 20 and 12 0 Hz). Add the sine waves and plot the sum. Then use a filter to remove signals above 50 Hz, and plot the filtered resHomework Equations The Attempt at a Solution I have attached an...
  19. S

    MHB The sine inverse of a purely complex number

    To prove that sin^{-1}(ix)=2n\pi\pm i log(\sqrt{1+x^2}+x) I can prove sin^{-1}(ix)=2n\pi+ i log(\sqrt{1+x^2}+x) but facing problem to prove sin^{-1}(ix)=2n\pi- i log(\sqrt{1+x^2}+x) Help please
  20. D

    EM wave described using a sine function.

    I'm going through my notes and I don't understand how they have included position in an equation to describe an EM wave. The equation is of the form http://upload.wikimedia.org/math/f/6/3/f6386c1751b91ec23c7123b15a11b52f.png [Asin(kx-ωt)]. This equation is just stated in my notes and there is no...
  21. A

    Protect SINE OUT BNC from SR830 lockin

    Hi, I am trying to use SR830 lock amplifier at scientific experintal. I need to apply the signal from SINE OUT BNC output in a thin ceramic sample. My sample is an plane disk with electrodes in opposite faces. This sample is isolante material, but sometimes the voltage can damage...
  22. D

    MHB Solve \(u_{tt} - u_{xx} = \sin(u)\): Let \(u(\xi)\)

    Consider \(u_{tt} - u_{xx} = \sin(u)\). Let \(u(\xi) = u(x - ct)\). How do we get \((1 - c^2)u_{\xi\xi} = \sin(u)\)? \[ u_{\xi\xi} = u_{xx} - c^2u_{tt} \] Correct?
  23. A

    Definite integration involving multiple angle of sine.

    pi ∫sin^2(nx)/sin^2(x) dx 0 I tried using mathematical induction and did arrive at the correct result but was wondering if a better method could be used to solve it?
  24. alyafey22

    MHB A generalized log sine integral .

    This thread will be dedicated to find a general formula for the integral I(a,t) = \int^t_0 x \log|\sin(a x )| \, dx \,\,\,\,\, a,t>0 This is not a tutorial . Any comments or attempts are always be welcomed .
  25. U

    Property of exponentials and sine functions?

    Homework Statement x''(t)+r*x't+kx=0 Suppose that for some initial conditions the solution is given by x=e^(-2t)*(3cos(t)+4sin(t)) What are are and k? Homework Equations See aboveThe Attempt at a Solution I've tried to "brute force" the solution simply by sticking the expression for x...
  26. A

    Can a half sine wave be inverted with a low-pass filter?

    I know that normal DC can be inverted to square wave , modified sine wave or pure sine wave , what about half sine wave dc , like the one resulted from ac current passing through a simple diode rectifier, can this be inverted?
  27. T

    Finding the Fourier Sine Series

    Find the Fourier SIne Series for f(x) = x on -L < x < L (Full Fourier) Ok, so my issue is in calculating the coefficients for the sine and cosine parts, more so an interpretation. So I have calulated the sine and cosine series to this point: let An: Cosine series Bn: sine series...
  28. T

    Intersections of Sine Graphs and Biorhythms

    Homework Statement Hi guys I was wondering how to find the points of intersections between 3 different sine graphs. For an assignment I am trying to find when my three biorhythms (Physical, Emotional, Intellectual) will all cross at once. Each cycle runs on the following time frame...
  29. S

    Exploring Sine Waves: Physical Shape vs. Graph Representation

    Is Sine Wave just a graph of particles that do SHM? A wave on a string is also a Sine wave provided its particles are moving sinusoidally. But its shape also kind of represents a sine wave. So is the wave on a string called a sine wave because of the physical shape that we can see or because...
  30. N

    MHB Longitudinal excitation of a sine wave

    Hi! This is a problem regarding a quarter car model driving over a sinusoidal road excitation. A sinusoidal excitation can be written on the form ze^(jωt), z being vertical. I would like to write the longitudinal excitation of a sine wave on the same form? Any hints and tips are much appreciated.
  31. H

    Using Fourier Sine basis to write x(L-x) [0,L]

    Homework Statement A function F(x) = x(L-x) between zero and L. Use the basis of the preceding problem to write this vector in terms of its components: F(x)= \sum_{n=1}^{\infty}\alpha _{n}\vec{e_{n}} If you take the result of using this basis and write the resulting function outside the...
  32. A

    Would a square wave look like a sine passed through a material?

    Sorry for the wording of the topic- I couldn't figure out how to make it fit. If I passed a hypothetical square wave through a material- be it wood, glass, cotton, etc.- would it change to look more like a sine wave?
  33. applestrudle

    Limits when there is a sine function?

    Homework Statement lim x-> ∞ xsin(1/x) Homework Equations The Attempt at a Solution I know that this is an ∞.0 type limit but I can't figure out how to change the sin function. Thank you
  34. dexterdev

    How to prove that a gaussian rv multiplied by a sine also guassian ?

    How to prove that a gaussian random variable multiplied by a deterministic sinusoid also results in a random variable with gaussian pdf? Suppose here A is a guassian random variable and B is given as below where fc is the frequency of sinusoid and t is the time. B=A∗cos(2π fc t). Is random...
  35. DreamWeaver

    MHB Evaluation of the Multiple Sine Function....

    Part 1:Define the Multiple Sine Function by \mathcal{S}_m(x)=\text{exp}\left(\frac{x^{m-1}}{m-1}\right) \prod_{k=1}^{\infty}\left(\mathcal{P}_m\left(\frac{x}{k}\right)\mathcal{P}_m\left(-\frac{x}{k}\right)^p\right)^q Where p=(-1)^{m-1}\, and q=k^{m-1}\, (these exponents weren't showing up so...
  36. G

    Circuit Problem with diodes and resistors with sine wave input

    Homework Statement Sketch Vo for the network shown (see attachment) and determine the dc voltage available. Homework Equations The Attempt at a Solution Vi is a sine like wave with amplitude of 100 V and goes on for one period. When Vi is positive, the diode on the right goes...
  37. MarkFL

    MHB Solve Calculus Limits w/ Sine Function: Answers to Hey's Questions

    Here are the questions: I have posted a link there to this topic so the OP can see my work.
  38. A

    Solve Fourier Sine Series - Get Help Here!

    Everything is in the picture I've attached. I believe my work is right because it's not that difficult of a problem but what I'm having a hard time seeing is how i go from the coefficient bn that I've calculated to the final solution. Maybe I did screw up or maybe there's an identity I'm not...
  39. Y

    Verify integration of a sine function

    I use part of HS-Scientist derivation in another post thanks to his detail derivation. I want to solving ## \int_0^{\pi} \sin(x sin(\theta)) d\theta## \int_0^{\pi} \sin^m(\theta) d\theta=-\frac{1}{m}\left[sin^{m-1}(x)cos(\theta)\right]_0^{\pi}+\frac{m-1}{m} \int sin^{m-2}(\theta) d\theta=...
  40. K

    Solve Tricky Sine Integral: $$\int_{0}^{\pi}\sin{n x}\sin{x}^3 dx$$

    Homework Statement Any ideas for how to solve the following integral? $$\int_{0}^{\pi}\sin{n x}\sin{x}^3 dx$$ where n is a positive integer Homework Equations Various sine and cosine identities The Attempt at a Solution I haven't much of a clue how to solve the integral...
  41. J

    I cannot make an arctan because there is a sine in the numerator

    \int\frac{2x^2sin(4x)}{1 + x^6}dx The solution should be substitution method... So far I've set u = x^3 , and made some progress trying to make the integral ready to become an arctan: \frac{2}{3}\int\frac{sin(4x)}{1 + u^2}du The set up would be fine if it were not for the sine term...
  42. Farmtalk

    MHB Prove Sum and Difference Sine?

    I was wondering if someone could take the time to show a proof for the sum and difference identity for sine. I've seen and learned to understand some other identities, but never this one. I've been trying to understand more of the "why" than the "how" of mathematics, and this one is very...
  43. J

    Derivative of Sine Function Raised to An Exponent

    Homework Statement Find the derivative of (sin x) ^ ((sin(sin x)))Homework Equations The Attempt at a Solution I get sin(sin x) * [(sin x) ^ {(sin(sinx)-1)}* cos x] The cos x isn't part of the exponent Is this right? Thanks :)
  44. Y

    Help with sine and cosine integral

    I am verifying the equation of radiation power of dipole antenna. I found mistakes in the derivation in the notes. I know the final equation is correct. So instead of following the steps in the notes, I reverse the step by using the final formula and going back step by step. Here is the final...
  45. P

    MHB How Do You Master Plotting and Sketching Sine Functions?

    Can anyone help me with this? I've been trying to figure it out but no luck. Where do I plot the points and how do I sketch the graph? All help is greatly appreciated!
  46. Y

    Series expansion of Sine Integral Si(x)

    This is not really a homework, I am trying to expand Si(x) into a series.The series expansion of Si(x) is given in articles: Si(x)=\int_0^x \frac{\sin\theta}{\theta}d\theta=\sum_0^{\infty}\frac {(-1)^k x^{2k+1}}{(2k+1)(2k+1)!} This is my work, I just cannot get the right answer: Si(x)=\int_0^x...
  47. Y

    What is the importance of studying Sine and Cosine Integral?

    I want to study a little bit more of Sine and Cosine Integral. I looked through all my textbooks including Calculus, ODE, PDE, Linear Algebra...Nothing! I found info on the web, but mostly are definitions. Where is this subject belongs to? Anyone can give me a link to a more complete...
  48. Petrus

    MHB Integrate Sine and Square root Composite Function

    Hello MHB, I got stuck on integrate this function \int \frac{\sin^3(\sqrt{x})}{\sqrt{x}}dx my first thinking was rewrite it as \int \frac{\sin^2(\sqrt{x})\sin(\sqrt{x})}{\sqrt{x}}dx then use the identity \cos^2(x)+\sin^2(x)=1 \ \therefore \sin^2x=1- \cos^2(x) \int...
  49. R

    Uncertainty with sine of angles?

    Homework Statement I'm in a Year 10 Physics class, and we have been doing an experiment about Snell's Law (θincident = θrefracted). The experimental design is fairly simple: a beam of light (from a ray box with a single slit in front of it) is shone into a glass block. The angles of the...
  50. R

    Does a sine-shaped beam remain sine when pushed?

    I have a linear elastic thin beam y=sin(pi x) from 0<x<1 and the beam is pin supported (no moment applied) at x=0 and x=1 and constrained so that the ends remain on the x axis. Then I push the end from x=1 to x=1-delta (for some small delta, say 0.1). Will the resulting beam shape still be a...
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