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. R

    Easy yes or no answer sine law question.

    Homework Statement \ Is it ok to use sine law for this : A=sIN^-1(40SIN120/65.8) im not sure if this applies to the sin law angle restriction where the angle can't be over ??degrees?
  2. J

    Help with 1kHz Sine Wave Setup for Mechanical System

    Hello, I need some help with the setup of a circuit. I am a mechanical engineering student and I am working on a mechanical system that employs the use of a sine wave at 1kHz, and the signal would need to be amplified to about 400-450V. It will be employed on a 24V DC system. I just need some...
  3. M

    Orthogonal Properties for Sine Don't Hold if Pi is involded?

    Orthogonal Properties for Sine Don't Hold if Pi is involded?? Normally I know \int_{-L}^L \sin \frac{n x}{L} \sin \frac{\m x}{L} ~ dx = 0\mbox{ if }n\not =m , \ =L \mbox{ if }n=m but apparently this doesn't work for \int_{-L}^L \sin \frac{\pi n x}{L} \sin \frac{\pi m x}{L} ~ dx I am...
  4. L

    Cant oscillate sine wave from bubba oscillator

    Cant oscillate sine wave from "bubba" oscillator hey i can't oscillate sine wave from "bubba" oscillator using pSpice. see the following attachment. can u please help me?
  5. Ƒ

    Integral of sine to an even power

    HOw do you integrate sin(x)^10?
  6. Z

    Image of upper half-plane under the inverse sine

    Homework Statement The problem is simply to find the image of the upper half-plane under the inverse sine function. Homework Equations The textbook defines the inverse sine in the following way. First, it defines arccos(w) = -i * log(w +/- sqrt(w^2 - 1)) and then it defines arcsin(w) = pi/2...
  7. M

    Sine Wave Addition: Standing Waves?

    If two sine waves have the same frequency and amplitude but have different phase shift do they still produce a standing wave? Thanks for the help.
  8. J

    I to do a Fourier Sine Transform

    Homework Statement A function u(x, t) satisfies the heat equation K\frac{\delta^{2}u}{\delta x^{2}} = \frac{\delta u}{\delta t} on the half line x \geq 0 for t > 0, where K is a positive constant. The initial condition is u(x, 0) = cxe^{\frac{-x^{2}}{4a^{2}}} with c and a being constants...
  9. E

    Solving Sine Fourier Series for f(x) = 1

    Homework Statement Find a sine Fourier series for the function f(x)=1 define on 0<x<1. use this series to show that \Sigma\stackrel{(-1)^k}{2k+1} =\stackrel{\pi}{4} betwen k=0 and infinity Homework Equations The Attempt at a Solution i found the Fourier series to be\Sigma...
  10. K

    Sine wave with variable frequency

    I'm trying to do something very simple... I'd like to have a sine function, where the frequency is controlled by a separate frequency function Something like this: g(t) = sin(2*pi*f(t)*t) Assume that f(t) = 20*exp(-2*t)+4 I would expect a sine wave that starts at 24 Hz and then...
  11. X

    Frequency Spectrum of a Real Sine Wave

    Hi. I'm doing a lab where we hooked up an RF signal generator at 1.25MHz/10dBm to a spectrum analyzer using a 50 ohm wire. Can anyone explain to me or link me to a place where I can read about why a real sine wave's frequency spectrum is not a pulse? Also, why would with a sweep from 1 Mhz...
  12. morrobay

    Y displacements in sine wave at (x) and (x + 2 wavelengths)

    Homework Statement A sine wave in + x direction with max amplitude of 1 wavelength= 2.85 m wavenumber, k = 6.28/2.85 m = 2.2 rad/m w=8rad/m frequency=1.27 cyc/sec vel.= 3.63 m/sec at x = 20m t= 5.5 sec at x + 2 wavelengths x= 25.7m t=7.07 sec by definition the y displacement is the...
  13. O

    Solving for a fixed point for a sine map

    Homework Statement Consider the sine map x{sub t+1} = f(x{sub t}) where f(x) = r*sin(x*pi). For r > 1/pi there are two fixed points, one at the origin that is unstable, and one elsewhere on the curve. The non-origin fixed point starts out, as you turn r just slightly above 1/pi, as stable...
  14. Phrak

    High-Precision 60 Hz Sine Wave Generator for Circuit Elements

    As a circuit element I need a 60 Hz sinewave with low harmonic distortion (maybe 1-2% max.) and precision amplitude (1-2% max) of about 2.5 volts. It should be syncronous with an input reference frequency. The reference is about 30KHz and is some as-yet-undetermined multiple of 60 Hz.
  15. H

    Calculating Forward and Backwards error of the sine function

    1. The sine function is given by the infinite series sin(x) = x - x3/3! + x5/5! + x7/7! + ... a) What are the forward and backward errors if we approximate the sine function by using only the first term in the series, for x = 0.1, 0.5, 1.0? b) Using the first two terms. Homework Equations...
  16. H

    Simple Sine lim Understanding question

    Homework Statement prove lim h-->0 sin(h)/h = 1 Homework Equations I understand the idea of squeze theorom. Understand that area of small triangle < sector area< big triangle. I know 1/2sin(h) < 1/2h < 1/2(sin(h)/cos(h)). In my calc book it goes from sinh< h< (sinh/cosh) to...
  17. J

    Proof of additive property for sine

    Homework Statement We are supposed to prove that sin(x+y) = cos(x)sin(y) + sin(x)cos(y) Homework Equations cos(A-pi/2) - sin(A) sin(pi/2 - A) = Cos(A) sin(A-pi/2) = -cos(A) The Attempt at a Solution We had to prove all of the relevant equations but were allowed to work in groups...
  18. S

    A sine curve coiled in a sinusoidal fashion?

    Ａ sine curve coiled in a sinusoidal fashion? Hi all, I am find the mathematical representation of a wire itself is of a sinusoidal shape, and now we coil this sinusoidal shaped wire into a sinusoidal shape again, so it's like a double sine wave. I have a rough idea, that is to first define a...
  19. C

    How to Prove Continuity of Sine Function at 0?

    Homework Statement Using the inequality |\sin(x)| < |x| for 0 < |x| < \frac{\pi}{2}, prove that the sine function is continuous at 0. Homework Equations Definition of continuity: A function f: R -> R is continuous at a point x0 \in R, if for any \epsilon > 0, there esists a...
  20. G

    Spring-damper system with sine input

    Homework Statement http://s3.amazonaws.com/answer-board-image/ad36bcba0e2d0b49d556f054de19d124.jpg Variable y(t) is the position of the block of mass m, and u(t) is the position of the plate on the right. The spring is unstretched when y = u. We can think of u(t) as the input and y(t) as...
  21. J

    RMS of square, sine and triangle waves

    I'm trying to calculate the RMS for square, triangle and sine waves. I can easily do the integrtion for sine waves and for square waves by looking at the graphic and getting the areas. It doesn't seem as easy for triangle waves since its squared form looks much more complicated and I'm not...
  22. H

    What is the Fourier Sine Transform of 1?

    Homework Statement I'm looking to determine the Fourier sine transfom of 1. Homework Equations One this site http://mechse.illinois.edu/research/dstn/teaching_files2/fouriertransforms.pdf (page 2) it gives the sine transform as \frac{2}{\pi \omega} The Attempt at a Solution...
  23. K

    How to Design a Pure Sine Wave Inverter for Motor Applications?

    I would like to design a Pure Sine wave inverter for DC to AC rectification. I want to drive motors with this applicaiton. (At most 120VAC .5HP) I don't want to specify too much, because the main point here is design basics. With the right information, I should be able to design accordingly. I...
  24. A

    Trig/constructing sine function from graph

    Homework Statement Construct or "work backwards" from a sine graph to a sine formula. The formula should be in: y = A sin(Bx+C) +D Where A is amplitude, D is Vertical Shift (on y access) and Bx+C are arguments of the function The graph shows that the Phase Shift is 1 and Period is...
  25. N

    Understanding the Heaviside Function and Rewriting Sine Homework

    Homework Statement f(t) = \left\{ \begin{array}{rcl} 5sin(t) & \mbox{for} & 0 < t < 2\pi \\ 0 & \mbox{for} & t > 2\pi \end{array}\right. Now, the problem is about rewriting f(t). My friend and I decided that it had to be \dfrac{10 - 5e^{-2\pi s}}{s^2 + 1} However, the answer turned out...
  26. K

    Sine series for cos(x) (FOURIER SERIES)

    I was finally able to figure out how to find the sine series for cos(x), but only for [0,2pi]. A question i have though is what is the interval of validity? is it only [0,pi]? Ie if I actually had to sketch the graph of the sum of the series, on all of R, would I have cosine or just a periodic...
  27. K

    Sine Integral Function in differential equation

    Homework Statement Si(x)=\int(sint/t)dt from 0 to x integrand is 1 at t=0 express the solutiony(x) of the initial value problem x^3y'+2x^2y=10sinx, y(0)=0 in terms of Si(x) Homework Equations The Attempt at a Solution y'+2/xy=10x^-3sinx multiply by x^2 x^2y'+2xy=10x^-2sinx...
  28. R

    How can I prove the sine theorem?

    Homework Statement How can I prove this theorem in the triangle ABC? sin(A)/a=sin(B)/b=sin(C)/c Homework Equations A*B and ... The Attempt at a Solution I have drawn a triangle and tried to prove it, but i couldn't. (I don't know how to send a picture to my post!) I know...
  29. A

    Trigonometric Identities for Sine and Cosine

    Homework Statement 4 \ sin \ \theta \ = \ 3 \ csc\ \theta The Attempt at a Solution sin\ \theta \ = \ \frac {3}{4} \ csc \ \theta sin^2 \ \theta \ = \ \frac {3}{4} sin \ \theta \ = \ \pm \ \frac {\sqrt{3}}{2} 30 \ \deg \ in \ QI, \ 150 \ \deg \ in \ QII, \ 210 \ \deg \ in \...
  30. P

    Why is Light in a Sine Wave Form?

    Why are light waves/X-rays/gamma rays/etc. in the form of sine waves, rather than, say, a zig zag wave, or even a straight line? I recently watched this youtube video explaining how to visualize ten dimensions: http://www.youtube.com/watch?v=JkxieS-6WuA&feature=related And wondered if photons...
  31. I

    Integration of 1/2 sin y dy from 0 to pi/2: Solution

    Homework Statement what is the integration of 1/2 sin y dy from 0 to pie/2 Homework Equations i know sin y = -cos y (integration) The Attempt at a Solution from 1/2 sin y dy to (1/2) -cos y then i plot pie/2 and 0 i got (1/2) [-cos (pie/2) - (-cos 0)] = 0, but the answer is 1/2...
  32. B

    Cosine and Sine rules to get magnitude and direction of a resultant force

    Use the cosine and sine rules to determine the magnitude and direction of the resultant of a force of 11 kN acting at an angle of 50 degrees to the horizontal and a force of 6 kN acting at an angle of -30 degrees to the horizontal. helppp please
  33. I

    How to Prove the Sine Rule Using Cross Product?

    Could anyone tell me how to use the cross product to prove the sine rule
  34. S

    Sine of Uniformly Distributed Random Variable

    Homework Statement Suppose U follows a uniform distribution on the interval (0, 2pi). Find the density of sin(U) Homework Equations The Attempt at a Solution Well if U ~ (0, 2pi), then sin(U) should follow a distribution on [-1, 1]. I know one way to do tackle such problems is to...
  35. L

    Derivative of a Sine Function Problem Confusion

    Homework Statement If f(x) = sin²(3-x), then f ' (0) = ____ . A. -2cos3 B. -2sin3cos3 C. 6cos3 D. 2sin3cos3 E. 6sin3cos3 Homework Equations Derivative of Sinx= cosx. The Attempt at a Solution I cannot seem to figure out what the derivative would be with this...
  36. H

    Partial of a Sine where the PHASE is the variable?

    Homework Statement For example: the function is A0sin(w0*t - B*z) If I take the partial derivative with respect to z, how do you go about this? In my years at uni, I don't know if this has ever come up. Homework Equations The Attempt at a Solution My initial thoughts are...
  37. N

    Propogation of error when taking sine inverse

    Homework Statement I need to calculate the angle of inclination of an air track. The hypotenuse is 229.8 +/- 0.05 cm and the opposite side (the height that one side of the track is raised to) is 1.3 +/- 0.05 cm. I need to calculate the error in the angle of inclination. Homework Equations...
  38. D

    Quick question about the range of a sine function

    Homework Statement I am doing a proof about limits, and I need to know the range that sin(n^3-2n)/n is within for all n. Homework Equations The Attempt at a Solution I know that sin(n) is in between -1 and 1 for all n, so sin(n)/n would be in between -1/n and 1/n for all n...
  39. E

    Is the product of two sine functions always real-valued?

    Hi, I was playing around with Euler's Identity, and I found something (or at least I think I found something) interesting: It is a well known identity sin(z) = [exp(iz) - exp(-iz)]/(2*i), where z is any complex number, exp is the complex exponential function, and i is the imaginary...
  40. E

    Calculate phase between sine waves given the time difference

    Homework Statement How do i find the phase between sine waves knowing the time difference between each? http://users.bigpond.net.au/exidez/sinewave.jpg Equation for the main function is 3cos(30t - Pi/2) The smaller sine wave is of the same frequency when is settles in a steady state...
  41. Y

    Expand the function f(x) = x^3 in a Fourier sine series

    Homework Statement Expand the function f(x) = x3 in a Fourier sine series on the interval 0 <= x <= 1 Homework Equations \[f\left( x \right)=\sum\limits_{k=1}^{\infty }{{{b}_{k}}\sin \left( k\pi x/a \right)}0<x\le a\] and \[{{b}_{k}}=\frac{2}{a}\int_{0}^{a}{f\left( x \right)\sin...
  42. O

    Build In-Phase Sine Wave Generator 1KHz-12KHz Voltage Controlled

    Help! I need to build an in-phase sine wave generator with quadrature output and its frequencies are between 1KHz-12 KHz and it need to be voltage controlled Thanks for any help anyone can provide.
  43. 0

    Signal that is a sine wave plus an offset

    I have a signal that is a sine wave plus an offset. I would like to measure the dc component and possibly also subtract it from the signal later. The signal is very slow (10th of Herz) and the offset changes on the order of minutes. I thought it should be possible to get a time average with an...
  44. B

    Linear interpolation of a sine wavetable

    Hi everyone. I'm doing a microcontroller project where I'm using a 256 element array of 1 byte values to output a sine wave at varying frequencies. I'm using the top 8 bytes of a 16 bit "angle increment" value that's incremented by varying amounts as an index to the array of values, to control...
  45. J

    Sine wave resonance using square wave input

    Hello All, I have been doing a few basic experiments with coils lately and I am trying to figure out the simplest way to drive a coil at its natural resonant frequency so that I can get the Q as high as possible. My only problem is that I don't have a sine wave generator that I can use. I...
  46. I

    Is a Sine Wave Always Transverse?

    hi all, pretty sure this is the wrong place to post this but it is a quick one so... is a sine wave always transverse? thanks
  47. G

    Sine wave to square/triangle wave.

    How are Square and Triangle waveforms produced from Sinusoidal waveforms? Thanks Greg
  48. D

    Newton's second law cosine and sine

    Newton's second law is taking my mind for a spin and for some reason had me contemplating how it works for several hours. This is all with respect to an incline and an object sitting on the incline with no friction. If the problem does not give you the mass of the object can you completely...
  49. P

    Power of a sine wave (electronics engineering)

    This is obviously an electronics question. In communication systems, to calculate the power of a sine wave, the formula below is used Power (Sine Wave) = 1/2 * (peak amplitude)^2 This formula is apparently a standard electronics formula. I'm trying to understand where it comes from. How...
  50. H

    Orthogonality of Sine and Cosine functions

    Hi, would anyone be able to explain how to evaluate a function using orthogonality (i.e. using orthogonality to solve a definite integration problem with sines/cosines)? Thank you
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