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. durandal

    Engineering Fourier transform of a shifted sine wave

    This is my attempt at a solution. I have used Eulers formula to rewrite the sine function and then used the Fourier transform of complex exponentials. My solution is not correct and I don't understand if I have approached this problem correctly. Please help. $$ \mathcal{F}\{\sin (4t-4) \} =...
  2. E

    What is the difference between the two sine rules for resultant vector?

    Hello can anyone help me with this: there are two sine rules for finding the direction of a resultant vector;one for the sides and one for the angle; I tested both formulas and they all worked well and gave me equal answers, does that mean I can use them interchangeably,the rules are: a/sinA =...
  3. T

    Why are the paths of our cosmic explorations, pretty?

    TL;DR Summary: Why are the paths of our cosmic explorations, pretty? OK, so I ask a lot of stupid questions. Here's another. Why is this picture, below, pretty? (They are the paths of all our cosmic explorations.) Now, I get the sine, cosine, circles, gravitational attraction, escape...
  4. Mr X

    Derivation or proof of derivative sin (x)

    How do I do this from here without using the derivatives of sin or cos ?
  5. K

    Can a Square Wave Tachometer Drive be Powered by a Sine Waveform?

    I'm a marine engine mechanic, and as engine controls & sensor systems have gotten more complicated with current technology, my shop gets more & more requests for instrumentation & control system repairs. I have a lot of trouble getting technical info from suppliers, so I have been starting to...
  6. A

    Is it possible to solve relative velocity problems without sine law?

    I was able to solve this question successfully by utilizing the sine and cosine law however my instructor said I was only allowed to utilize the vector component method, I am unsure how to complete this question using the vector component method as we have two unknowns(those being the angle of...
  7. A

    Fourier sine and cosine transforms of Heaviside function

    Hi, I am really struggling with the following problem on the Fourier sine and cosine transforms of the Heaviside unit step function. The definitions I have been using are provided below. I tried each part of the problem, but I'm only left in terms of limits as x -> infinity of sin or cos...
  8. S

    Find limit involving square of sine

    $$\lim_{n \rightarrow \infty} \sin^{2} (\pi \sqrt{n^2+n})$$ $$=\lim_{n \rightarrow \infty} \sin^{2} (\pi \sqrt{n^2+n}-n\pi)$$ $$=\lim_{n \rightarrow \infty} \sin^{2} (\pi \sqrt{n^2+n}-n\pi)$$ $$=\lim_{n \rightarrow \infty} \sin^{2} (\pi (\sqrt{n^2+n}-n))$$ $$=\lim_{n \rightarrow \infty} \sin^{2}...
  9. Purplepixie

    MHB Closed form solution to sum of sine positive zero-crossings

    Hello, I would like to know, if there's a closed form solution to the following problem: Given a sum of say, 3 sines, with the form y = sin(a.2.PI.t) + sin(b.2.PI.t) + sin(c.2.PI.t) where a,b,c are constants and PI = 3.141592654 and the periods in the expression are multiplication signs, what...
  10. jaumzaum

    I Why does the integral of sine of x^2 from - infinity to + infinity diverge?

    Hello guys. I was trying to evaluate the integral of sine of x^2 from - infinity to + infinity and ran into some inconsistencies. I know this integral converges to sqrt(pi/2). Can someone help me to figure out why I am getting a divergent answer? $$ I = \int_{-\infty}^{+\infty} sin(x^2) dx =...
  11. George Keeling

    B Happy Christmas: Is it Really True?

    Is this really true? It resembles the binomial theorem. I've posted it twice which might be breaking rules. Happy Christmas PF.
  12. B

    I Equation to graph a sine wave that acts like a point on a unit circle

    I need an equation to graph a sine wave that act like a unit circle but only positive numbers. so I need it to be 0 at 0, A at 90 , 0 at 180, A at 270, 0 at 360, and A at 450 and so on and so on... Now I know sin(0) is 0 in degrees and sin(90) 1 and I know if you Square a number is...
  13. S

    I How Do You Calculate Day-Specific Phase Shifts in Excel for Sine Waves?

    Hi, I have created a sine wave with the following options: 1.) - changing the period/length in days of the sine wave (Cycle Length in Days) 2.) - calculating the start value of the "dummy" so that the sine wave always starts with -1 (Dummy Start at Cycle Trough) when the phase shift is set...
  14. tworitdash

    A Fourier Transform of an exponential function with sine modulation

    I want to know the frequency domain spectrum of an exponential which is modulated with a sine function that is changing with time. The time-domain form is, s(t) = e^{j \frac{4\pi}{\lambda} \mu \frac{\sin(\Omega t)}{\Omega}} Here, \mu , \Omega and \lambda are constants. A quick...
  15. MountEvariste

    MHB Definite integral involving sine and hyperbolic sine

    Calculate $\displaystyle \int_0^{\infty} \frac{\sin x}{\cos x + \cosh x}\, \mathrm dx.$
  16. J

    B RC Low Pass Circuit Sine Wave Response

    Hey Everyone, I am trying to gain a level of fundamental understanding of an RC circuit sine wave response through the mathematics and was wondering if someone could help me work it out. Fundamentally a sine wave is represented by the equation y=-ky'' . When a sine wave is used as the input...
  17. anemone

    MHB Linear combination of sine and cosine function

    Hi MHB! I recently came across a problem and I was thinking most likely I was missing something very obvious because I couldn't make sense of what was being asked, and I so wish to know what exactly that I failed to relate. Question: Find the minimum of $6\sin x+8\cos x+5$. Hence, find the...
  18. anemone

    MHB Trigonometric of tangent and sine functions

    Simplify $\left(\tan \dfrac{2\pi}{7}-4\sin \dfrac{\pi}{7}\right)\left(\tan \dfrac{3\pi}{7}-4\sin \dfrac{2\pi}{7}\right)\left(\tan \dfrac{6\pi}{7}-4\sin \dfrac{3\pi}{7}\right)$.
  19. penroseandpaper

    Odd and even extension of sine function

    Hi everyone, We've been looking at Fourier series and related topics in online class, touching upon odd and even periodic extensions. However, we haven't looked at what this translates to for sine and cosine functions - only sawtooth and line examples. So, I'm trying to do my own investigation...
  20. C

    Pure Sine Wave Inverter using Two Lithium Ion Batteries

  21. anemone

    MHB Can You Meet the Sine Function Challenge?

    Let $a,\,b$ and $c$ be real numbers such that $\sin a+\sin b+\sin c\ge \dfrac{3}{2}$. Prove that $\sin \left(a-\dfrac{\pi}{6}\right)+\sin \left(b-\dfrac{\pi}{6}\right)+\sin \left(c-\dfrac{\pi}{6}\right)\ge 0$.
  22. Theia

    MHB Computing the value of sine function accurately

    Hi all Simple question: How I can compute the value of \(a = \sin \left( 2017 \sqrt[5]{2} \right) \) under following assumptions: No use of advanced numerical libraries is allowed. Only accepted operations are: comparisons, absolute value, addition, subtraction, multiplication and division...
  23. bob012345

    I Exact Expression for Sine of 1 Degree

    For example, $$Sin(15)= \frac {(\sqrt 6 - \sqrt 2)} 4$$ and $$Sin(3)=\frac {(\sqrt 6 (\sqrt 5 -1)(3+\sqrt 3)} {48} -\frac {\sqrt 3 (3-\sqrt 3 )\sqrt{ 5+\sqrt 5 }} {24}$$ What about ##Sin(1)##?
  24. N

    A Sine Dipole formation using two hydrophones

    Hi all. I am trying to do MATLAB simulations for generation of SINE Dipole using two hydrophones spaced distance 'd' apart for signal coming from direction 'DOA'. The MATLAB code is given below. The confusion is that there is constant phase difference of 90 degree b/w SINE Dipole generated using...
  25. F

    I Determining the Equation of a Sine and Cosine Graph that speeds up

    My function needs to speed up towards the left. How do I do this? Green is the graph. Red is my function that needs to match the graph. A = Amplitude = -0.13 H = Phase Shift = 0.1625 V = Vertical Shift = 0.05 P = period = 0.4 B = 2π / P Y = A (Cos(B...
  26. S

    B What is the speed of a photon traveling along the sine function?

    On the image you can see a photon starting at point A at t=0. The photons travels along the sine function and arrives point C. I knot that this takes T=λ/c. But this is the time for a object traveling directly from the origin to point C and not along the sine wave! If the photon travels...
  27. bagasme

    B Derivation of Cosine and Sine Method of Vector Sum

    Hello all, In high school physics, the magnitude sum of vector addition can be found by cosine rule: $$\vec {R^2} = \vec {F^2_1} + \vec {F^2_2} + 2 \cdot \vec F_1 \cdot \vec F_2 \cdot cos ~ \alpha$$ and its angle are calculated by sine rule: $$\frac {\vec R} {sin ~ \alpha} = \frac {\vec F_1}...
  28. arcTomato

    Engineering The power spectrum of a sine wave (C language)

    Hi I would like to Derive the power spectrum of sinusoid.I tried like this. But It doesn't work. <Moderator: CODE tags added> #include <stdio.h> #include <math.h> #define pi 3.1415926535 FILE *in_file, *out_file; int main() { dft(); } int dft(int argc, char *argv[]) { char...
  29. Kirkkh

    B Why is sine not used for dot product?

    There’s a old 2012 post on here “Why sine is used for cross product and cosine for dot product?” —there are a lot of great answers (which is how I came about this forum). After reading over the replies, it occurred to me: really it’s just because cosine is the “start” of a unit circle. Which...
  30. A

    I How can I go from sine to cosine using exponential numbers?

    ##cos(\omega)## is $$\frac{e^{j \omega } + e^{-j \omega }}{2}$$ ##sin(\omega)## is $$\frac{e^{j \omega } - e^{-j \omega }}{2 j}$$ I also know that ##cos(\omega - \pi / 2) = sin(\omega)##. I've been trying to show this using exponentials, but I can't seem to manipulate one form into the...
  31. I

    A How can you generate a sine wave using integers only?

    I need to recursively generate a quadrature signal which fits exactly into a grid NxN, where N is a large power of two. After unsuccessful research, I decided to develop my own solution, starting from the waveguide-form oscillator taken from Pete Symons' book 'Digital wave generation, p. 100'...
  32. karush

    MHB 2.2.206 AP Calculus Practice question derivative of a composite sine

    206 (day of year number) If $f(x)=\sin{(\ln{(2x)})}$, then $f'(x)=$ (A) $\dfrac{\sin{(\ln{(2x)}}}{2x}$ (B) $\dfrac{\cos{(\ln{(2x)}}}{x}$ (C) $\dfrac{\cos{(\ln{(2x)}}}{2x}$ (D) $\cos{\left(\dfrac{1}{2x}\right)}$ Ok W|A returned (B) $\dfrac{\cos{(\ln{(2x)}}}{x}$ but I didn't understand why...
  33. Robin04

    Asymptotic expansions of the sine function

    There are no restrictions for ##a,b,f_1,f_2##. One solution is the first order Taylor series expansion of course with ##f_1(a)=a,f_2(b)=b##, but are there any other solutions? I tried the Bhaskara formula but I couldn't express it in this form.
  34. Benjamin_harsh

    How is the Sine law written for this problem?

    Find the resultant vector of vectors A and B shown in the figure. Solution: By geometry method: Cosine law for the right side triangle. ##R^{2} = 17^{2} + 44^{2} - 2 (17)(14).cos 70^{0}## ##R = 41.39 m/sec## By Sin law, ##\large\frac {R}{Sin 70^0} = \frac {17}{Sin\alpha}## ##sin...
  35. J

    I Are electromagnetic waves sine waves?

    Light is said to consist of photons or electromagnetic waves. I'm not asking which view is correct, what conditions make one view or the other more useful, or advantages and disadvantages of each view. I am assuming the two views are compatible to the extent that the wave character of light can...
  36. Benhur

    Combining Sine Functions: Simplifying with Trigonometry

    Moved from technical forum, so no template is shown Summary: I have the expression sin(2x) + sin(2[x + π/3]) and I have to write this in terms of a single function (a single harmonic, rather saying). But I don't know how to do this, and... it seems a little bit weird for me, because I'm merging...
  37. M

    Uniform convergence of a sine series

    I'm not too sure how to use the hint here. What I had so far was this: an odd extension of ##f## implies ##f = \sum_{k=1}^\infty b_k \sin(k x)##. Notice for ##m>n## $$ \left|\sum_{k=1}^m b_k\sin(k x) - \sum_{k=1}^n b_k\sin(k x)\right| = \left| \sum_{k=n+1}^m b_k\sin(k x)\right| \leq...
  38. FQVBSina_Jesse

    A Bessel's Integrals with Cosine or Sine?

    Hello all, This is knowledge needed to solve my take-home final exam but I just want to ask about the definition of Bessel's integrals. This is not a problem on the exam. Wikipedia says the integral is defined as: $$J_n(x) = \frac {1} {2\pi} \int_{-\pi}^{\pi} e^{i(xsin(\theta) - n\theta)} \...
  39. Morbidly_Green

    Finding the Sine Representation of an Odd Function Using Fourier Series

    I am attempting to find the sine representation of cos 2x by letting $$f(x) = \cos2x, x>0$$ and $$-\cos2x, x<0$$ Which is an odd function. Hence using $$b_n = \dfrac{2}{l} \int^\pi _0 f(x) \sin(\dfrac{n\pi x}{l})dx$$ I obtain $$b_n = \dfrac{2n}{\pi} \left( \dfrac{(-1)^n - 1}{4-n^2} \right)$$...
  40. E

    Line voltage in 3 phase as single sine wave

    Howdy all. The typical image of a three phase electrical system involves 3 sine waves, phase shifted 120 degrees. These sine waves each, individually, represent the 'phase voltage,' which is to a common neutral in a wye configuration. In this wye configuration the line to line voltage is...
  41. Miles123K

    The sum of this series of the product of 2 sine functions

    Homework Statement I have encountered this problem from the book The Physics of Waves and in the end of chapter six, it asks me to prove the following identity as part of the operation to prove that as the limit of ##W## tends to infinity, the series becomes an integral. The series involved is...
  42. E

    MHB Express in terms of sine function of f(x)=sinx+cosx

    f(x)=sinx+cosx getting really frustrated with my math teacher. gives us forumlas for things but then barely shows us how to use them if at all and then throws problems at that we have to make sense of ourself. why can't math teachers teach? anyway, the question is express f(x)=sinx+cosx in...
  43. Matt Benesi

    B What are cosine and sine functions called in relation to Pi?

    1)* What are sine and cosine functions called in relation to Pi? 2) What is the exponential function called in relation to cosine and sine functions? 3) What are the other smooth, continual nested (or iterative) root functions (that are similar to sine and cosine) called in relation to...
  44. V

    B How to set up an integral to integrate over a sine wave?

    How do I setup an integral to integrate over the following equation: V(t) = 1/(R*C) integral to t Vin(t) dt This is the capacitor voltage over time formula. I want to integrate over a sine wave from 9 to 81 degrees. Frequency of 120Hz, amplitude of 120V. The formula I used in wolframalpha is...
  45. Krushnaraj Pandya

    Is the Sine Rule Valid for Non-Triangular Vector Configurations?

    Homework Statement If magnitudes of vector a,b and c are 1,2,3 respectively and vectors a+b+c=0 then it is obvious that a and b will be in the opposite direction to c therefore cancelling it out but let's assume for a moment that we don't know this and we want to figure this out using the sine...
  46. Saracen Rue

    B Area under a sine integral graph

    Hello, I've recently discovered the sine integral and have been playing around with it a bit on some graphing software. I looked at the graph of ##Si(x^2) - \frac π 2## and saw that both the amplitude and period was decreasing as x increased. Curiosity got the best of me so I decided to...
  47. opus

    B Cosine or Sine of (angle+angle) always equal to 1?

    I'll start off with a given problem. Find ##cos\left(α-β\right)## given that ##cos\left(α\right)=\frac{-12}{13}## and α is in quadrant III. ##sin\left(β\right)=\frac{-5}{13}## and β is in quadrant III.Solution: ##cos\left(α-β\right)=1## This had we wonder if this continued for other angles of...
  48. opus

    B Adding Sine and Cosine Waves- How to get formula

    I have included a screenshot of a part of my textbook that is giving me a slight bit of confusion. It's talking about how to get the formula for adding sines and cosines. The part that I am confused about is the very first formula introduced in the screenshot. From what I understand, we are...
  49. T

    MHB Integral of sine = 27/2+ln^2(2)+ln(2)

    How to prove this integral, $$\int_{0}^{2\pi}\sin\left({x\over 2}\right)\ln^2\left[\sin\left({x\over 4}\right)\sin\left({x\over 8}\right)\right]={27\over 2}+\ln^2(2)+\ln(2)$$
  50. D

    What is the mistake in calculating the integral of the absolute sine function?

    Homework Statement \int_0^{2018 \pi} \lvert \sin(2018x) \lvert \mbox{d}x Homework EquationsThe Attempt at a Solution So the period is: \frac{2 \pi}{ 2018} Each "hump" of the sine has an area of 2 so if I count the number of humps I am done. In one period of an absolute sine function the...
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