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. S

    Finding the Intersection of a Sinusoidal Function and a Line

    Homework Statement Hi! I'm trying to find the points of intersection of a sinusoidal function and a line. The line is y=x/7. The function is y=sinx. Can someone tell me how to determine the number of intersections and exact intersections. I would also like to know if the same method can be...
  2. M

    Polygon sine functions? what is this?

    Hi! I was wondering how I could find the equations for the bottom two functions. I understand that the amplitude is not constant like that in the circular sine function--could someone please help me out? Thanks!
  3. K

    Envelope function for two sine curves

    Homework Statement Hi,[/B] I am trying to get an envelope function for two sinusoidal curves (A, B) added up. a1,a2 are the amplitudes(metre), T1,T2 are the periods (hrs), B lags by dt from A at t=0. case-1: a1= 1, a2=0.5, p1=11,p2=10,dt=0 w1=2*pi/T1,w2=2*pi/T2 A = a1*cos(w1*t)...
  4. L

    Work & Energy: Forces with Angles

    Homework Statement A student could either push or pull, at an angle of 30 degrees from the horizontal, a 40kg crate, where the coefficient of kinetic friction is .21. The crate is moved 18m. Calculate the minimum work for pushing and pulling. Homework Equations W=F•(change in)X•cos(angle in...
  5. Matterwave

    Finding the sine of the angle between 2 rays

    Hello, say you are on the unit sphere and you have 2 rays intersecting it from the origin. You know the spherical coordinates of where these 2 rays intersect the sphere ##(\theta_1,\phi_1),(\theta_2,\phi_2)##. Now, because we know the dot product of two vectors, it is simple to get that the...
  6. A

    How Does Offset Impact RMS Voltage Calculation?

    Homework Statement The rms voltage for a sine wave with zero offset is given by Vrms= 1/(√2)Vpeak. Calculate the rms voltage for a sine wave with a peak-to-peak voltage of 1V. Homework Equations Vpeak-to-peak=2Vpeak The Attempt at a Solution This may be a really easy question with an answer...
  7. T

    Can fourier sine series approximate even functions?

    I am learning Fourier series and have come across the sine, cosine, and imaginary exponential expressions. To my knowledge, these individual terms form a basis since they are all orthogonal to each other. I am just wondering: can a Fourier sine series be used to model a purely even function...
  8. S

    What is the frequency of the sum of several sine waves?

    I am given three sine waves with individual frequency being 10 Hz, 50 Hz, and 100 Hz. What is the frequency of the following : y(t) = sin(2π10t) + sin(2π50t) + sin(2π100t) Is it simply 100, the LCM of all the sin waves? If not, How to calculate the frequency of y(t) ?
  9. Tesladude

    Calculating Wattage with Sine Wave

    So my question is about calculating wattage with a sine wave. So with speakers I always just thought of it as the basic vi=w So a sine wave of 12v through 4 ohms will produce three amps and thus 36watts. But when you put a square wave and use pulse width modulation on something like an led or...
  10. Stephanus

    Understanding the Limits and Derivatives of Sine and Cosine Functions

    Dear PF Forum, In previous threads, I have asked about sine and cosine. The answer given by the members/mentors/advisor are very clear. But lengthy. Perhaps these yes/no questions that I can simply remember and not forget it (again). So here we are 1. if h = 0 then sin(h) = 0 2. if ##\lim_{h...
  11. Stephanus

    Proof limit derivatave of sin(x)

    Dear PF Forum, Continuing our debate discussion in differential in slice of X. I read this particulare website. About proofing the derivative of sine(x). http://tutorial.math.lamar.edu/Classes/CalcI/ProofTrigDeriv.aspx In there, the web writes arc AC < |AB| + |BC| < |AB| + |BD|...
  12. CupOfAppleTea

    Intro Physics Books on Phase and Amplitude of a sine wave

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  13. Orange-Juice

    Is the sine function defined at pi/2?

    Wouldn't the sine function be undefined at pi/2 since at that point there would be no triangle in the unit circle, only a straight line along the y-axis? The hypotenuse of a right triangle must always be the longest side of the triangle so I don't see how the sine function can ever give you a...
  14. D

    Discrete Fourier Transform of Sine Function

    (1) For a real function, g(x), the Fourier integral transform is defined by g(x) = \int_{0}^{\infty} A(\omega )cos(2\pi \omega x)d\omega - \int_{0}^{\infty} B(\omega )sin(2\pi \omega x)d\omega where A(\omega ) = 2 \int_{-\infty}^{\infty} g(x)cos(2\pi \omega x)dx and B(\omega ) = 2...
  15. C

    How was the Sine function programmed? The actual equation?

    Hi, I understand that Sine (angle) = opposite/Hypothenus is the ratio of the length of 2 sides in the triangle. However something i never understood and REALLY want to understand is. When you type in some angle in degree in a calculator like sin(14,123221) the calculator spits out a ratio that...
  16. M

    What Is the Amplitude of a Harmonic Load in the Frequency Domain?

    Hello, I am having a bit of trouble with calculating the Fourier transformation of a harmonic load. I have the function f(t) = A * sin(ωt) in the time-domain. I would like to represent this function in the frequency domain. What would be its amplitude? Thank you
  17. M

    Why do we have both sine and cosine?

    This might seem like a really basic question that one might cover in gr 9 or 10 but instead my friend and I were discussing it now, when he just got his degree and I'm a credit away from mine: why on Earth are there both sine and cosine functions when simply one would do? Either can be...
  18. L

    Question about sine values for angles 0-180

    So my math teacher told me that as long as an angle is less than 90 degrees. The sine value of that angle will also be equal to 180 subtracted by that angle. Why is it this way? I just don't understand why it is true spesifically for sine values. Is there are a way to compare this to triangles...
  19. D

    Fourier Transform of a full rectified sine wave

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  20. Q

    Calculating Infinite Sine Sum with Ratio Test | x and t Real Numbers

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  21. D

    Sine wave relationship with physical waves

    This is probably pretty basic but I've never actually seen an explanation of how exactly the sine wave relates to the physical waves it is so commonly used to represent. Could it be imagined as like the periodic thumping of a speaker where the peak of the sine graph represents maximum air...
  22. I

    How Did the Author Determine the Fourier Sine Series for x^2?

    I'm having trouble understanding a part in my book. second to last paragraph where it says 4.2 must be the Fourier sine series for x^2, how did the author arrive at that? http://i.imgur.com/gLLUYXw.jpg
  23. E

    How do I represent this sine function?

    I was struggling to represent the following for integer values of n: \sin \left( \dfrac {n\pi } {2}\right) I know for even n, we get zero But for odd n, it alternates beween 1 and -1 for every other odd. Is there a compact way to represent that? I feel like I'm being dumb and missing...
  24. H

    Square wave and sine wave -- How standing waves are formed?

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  25. G

    Fundamental frequencies of square wave and sine wave

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  26. A

    MHB Solving Sine Rule Question: Angle B & A in Triangle ABC

    I understand how to use the Sine rule, but I think I may get stuck halfway through! My questions is: In a Triangle ABC the angle at C is 48.15 degrees, side AB is 15.3m and side AC is 17.6m. Calculate the size of angle B and angle A, giving all possible solutions, in degrees accurate to 1 d.p...
  27. J

    Fourier sine series integration

    Homework Statement The question is to get Fourier sine series of e^-x =f(x) on 0<x<1 Homework Equations Bn = 2/L ∫ (e^-x) * sin(nπx/L) over the limits 1 to 0, where L = 1 f(x) = summation of Bn*sin(nπx/L) The Attempt at a Solution So I integrated ∫ by part integration so I took u =...
  28. N

    Are waves always the sum of sine waves?

    I saw that somewhere and it is supposed to be something Fourier came up with but I can't find somewhere why... Please explain (with mathematical description if possible)
  29. N

    Deriving the formula for sine waves

    Can anybody out there show me how the sine wave formula y=Acos(kx - ωt + φ_{0}) or y=Acos(kx + ωt + φ_{0}) is the direct solution of the wave equation \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2} ? I mean I looked it over on the Internet but everybody keeps...
  30. Y

    MHB What is the limit of the sine function as x approaches 1?

    Hello all, I am looking for the limit of: sin(x-1) / (x-1) where x approaches 1. How do I do that ? Thanks !
  31. P

    What Mistakes Are in My Integral Calculations?

    Find the value of the integral: a) ∫0π(sinx + 2)dx Formula I found: sin x dx = -cos x + C My calculation: F(x) = -cosx + 2x => (-cosπ + 2π)-(-cos0) = -1 + 2π + 1 = 2π , but the solution should be 2π +2 b) ∫02πsin(x/2)dx My calculation: F(x) = -cosx/2 => -cosπ + cos0 = 0 ; but the solution...
  32. H

    Curious definite integral : sine integral times exponential

    Hello PF, I just found a curious integral. I wondered if it comes from a bigger group of integral definitions: \int_0^\infty \mathrm{Si}(ax)e^{-x}\mathrm{d}x=\mathrm{atan}(a) Where Si(x) is the sine integral function \mathrm{Si}(x)=\int_0^x \frac{\mathrm{sin}x}{x}\mathrm{d}x I proved the...
  33. H

    Curious definite integral : sine integral times exponential

    Hello, and thanks for welcoming me in the forum of Physics Forums. I just found a curious integral that I solved by Taylor series. I wondered if it comes from a bigger group of integral definitions: ##\int_0^\infty \mathrm{Si}(ax)e^{-x}\mathrm{d}x=\mathrm{atan}(a)## Where Si(x) is the sine...
  34. C

    Sine function, need some help understanding its meaning.

    Hi when we look at the function Sin(35) = opposite/ hypotenus I know i can find the opposite side by Sin(35 degree) * hypotenus But the value of sin(35 degree) = 0,6293 so my question is this: The sine function tells us that if we divide opposite side by hypotenus the ratio is 0.6292:1...
  35. P

    Find the fourier sine series of cosine.

    Homework Statement Hi, so I am doing some past exam papers and there was this question; Homework EquationsThe Attempt at a Solution a0 and an both are equal to zero, this leaves only bn. Since you can only use the sine series for an odd function, and cos(t) is even, does this mean i have to...
  36. A

    MHB Prove $\displaystyle \lim_{x \to 0}\frac{x}{1 + \sin^2(x)} = 0$

    Hello: Prove $\displaystyle \lim_{x \to 0} \frac{x}{1 + \sin^2(x)} - 0$ Let $|x| < 1 \implies -1 < x < 1$ $\sin^2(-1) + 1 < \sin^2(x) + 1 <\sin^2(1) + 2$ $\implies \displaystyle \frac{1}{\sin^2(-1) + 1} > \frac{1}{\sin^2(x) + 1} > \frac{1}{\sin^2(1) + 1}$ $\implies \displaystyle...
  37. A

    Can anybody check this proof for a Sine limit?

    Mod note: Fixed the LaTeX. The closing itex tag should be /itex, not \itex (in brackets). I find it easier to use # # in place of itex, or $ $ in place of tex (without the extra space). Homework Statement Prove \lim_{x \to 0} \frac{x}{\sin^2(x) + 1} = 0 Homework Equations Given below: The...
  38. D

    ODE with base excitation caused by a half sine wave

    Homework Statement The suspension system of a car traveling on a bumpy road has a stiffness of ##k = 5\times 10^6## N/m and the effective mass of the car on the suspension is ##m = 750## kg. The road bumps can be considered to be periodic half sine waves with period ##\tau##. Determine the...
  39. J

    MHB How to Calculate $\int_0 ^{\infty} \sin t^2$ Using Complex Analysis?

    Hello. I'm having trouble calculating $\int_0 ^{\infty} \sin t^2$ using the fact that $\int _{\partial Tr} e^{-z^2} dz = 0$, where $Tr = conv (\{ 0, r, r + ir \})$ (a triangle). I'm aware that I need to somehow transform $e^{-z^2}$ to get $\sin t^2, \ \cos t^2$ but I don't know how to do...
  40. V

    Understanding Analog Signals & AC

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  41. S

    Create a Perfect Signal with Sine Wave Summation

    Hi I just wanted to check my approach. I have spectrum I have peak at 10Hz another at 20Hz and a third at 30Hz. The amplitudes are 1000, 500, 250. I want to recreate the signal by summing sine waves. I assume that I will therefore take A1 = 1; A2 = 0.5; A3 = 0.25; I will then let y =...
  42. S

    Integrating sine where argument goes to infinity.

    After some integration, i am getting a form e^{i \alpha\phi+i\beta\phi\sin(\phi-\phi')-i\gamma\sin\phi} , where ##\alpha, \beta, \gamma## are constants. Now i want to apply the limit where ##\phi ## ranges from 0 to ##\infty ## (ya, in the argument of sine we will encounter ##\infty ## which is...
  43. P

    What is the true frequency of the uncalibrated generator?

    Imagine that you have a calibrated sine wave generator set at 900 Hz and you mix its signal with that of an uncalibrated generator also set at 900 Hz. You hear a beat frequency of 4 Hz. What is the true frequency of the uncalibrated generator? Is there more than one possibility? I'm...
  44. T

    Exploring Complex Roots of Underdamped Systems: Why the Sine Term?

    I'm trying to do some refreshing of differential equations featuring damped systems. Specifically, I have a question regarding the differential equation solution to an under damped system involving complex roots. Referring to the attached pdf, an under damped system will yield a complex...
  45. KleZMeR

    Separate Sine Function with Complex Numbers

    Homework Statement I'd like to separate this function to U(x) + i*V(y) form. It's a homework problem that is asking if it is an analytic function. Searching thru trig substitutions, but looking ahead I don't see much luck... Any suggestions or help is greatly appreciated. Homework...
  46. 8

    Expand f(x)=x^3 in Fourier Sine Series: Step by Step Guide

    1. Expand the function f(x)=x^3 in a Fourier sine series on the inteval 0≤ x ≤ 1 2. I was thinking of using these equations in an attempt to find the solution f(x)=∑b_{n}sin(nx) and b_n=\frac{2}{∏}∫f(x)sin(nx)dx where n=1,2,...,I am somewhat lost in what to do exactly, could anyone help...
  47. T

    Sine Graph Help: Calculating Uoutput & Drawing Graph

    Homework Statement TASK So I have this task with opamp and I have Uinput Sine Graph, now I have to calculate Uoutput and write it as sine function and draw the sine graph. I know that this is an inverting OPAMP which means that output signal will be inverted, so I just have to flip this...
  48. G

    If Integral with Sine Limits What is Second Derivative?

    Homework Statement If f(x) = ∫sin x0 √(1+t2)dt and g(y) = ∫3y f(x)dx, find g''(pi/6)? Homework Equations FTC: F(x) = ∫f(x)dx ∫ab f(t)dt = F(b) - F(a) Chain Rule: f(x) = g(h(x)) f'(x) = g'(h(x))h'(x)The Attempt at a Solution I tried u-substition setting u = tan(x) for the first dirivative...
  49. E

    Dickey Fuller Test Sine Wave or seasonallity

    Hi can someone please clarify info regarding the application and validity of the Dickey Fuller test. If I perform the test using a sine wave would I be required to somehow take into account the seasonality. From my understanding a sine wave is non stationary, using the MATLAB command below I...
  50. C

    Series expansion of the signum function of sine

    Hi, I just read a physics paper and there it expands signum of a sine function as below: sgn(sin(wt))=(4/pi)*{sin(wt)+(1/3)*sin(3wt)+(1/5)*sin(5wt)+...} How can we expand sgn(sin(wt)) like this? Thanks.
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