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

    Understanding when to use sine and cosine to find x and y components

    I am having trouble understanding when to use sine and cosine to find x and y components. I know that its not always going to be the same (ex. you won't always use cosine to find x component.) Any input would be appreciated!
  2. M

    What is a wavefront? sine or cosine graph?

    What is a wavefront? Huygen's principle says that every point on a wavefront acts as a source of wave with the same speed. Now I am not asking you what that means, but i don't understand what a wavefront is. Is that like a circle? But wave is usually like a sine or cosine graph, so what's the...
  3. W

    Geometrical interpretation of Taylor series for sine and cosine?

    I've stumbled upon what might be a geometrical interpretation of Taylor's series for sine and cosine. Instead of deriving the Taylor's series by summing infinite derivatives over factorials, I can derive the same approximation from purely geometrical constructs. I'm wondering if something...
  4. L

    Is a gaussian distribution 'like a sine wave'

    So I was having a conversation with the guy I share an office with and I brought up the gaussian distribution to show the probability distribution of energies of electrons generated by a filament. He mentioned that it 'looks like a sine wave', and I said 'sorta, but it's not a sine wave'. He...
  5. I

    Solving the Mod of Sine Equation: What Does Each Number Mean?

    Homework Statement Guy records data of person using a swing. Comes up with an equation to show his findings: y = 1.6sin(1.8x - 0.9) + 2.5 What does each number represent in relation to the situation (person swinging)? Homework Equations Don't think any equations are needed for...
  6. E

    Creating Sine Wave - Oscillator/Chips for 100kHz Output

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  7. B

    What is the expression for a series of sine equations in a spring-mass system?

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  8. T

    How do I solve this odd power of sine integral problem?

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  9. D

    Mastering Trigonometry: Understanding Sine, Cosine, and Tan Graphs

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  10. L

    Area enclosed by sine and cosine.

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  11. R

    What are the period and phase shift in the function y=4sin(3X+Pie/4)-2?

    In the function y=4sin(3X+Pie/4)-2 what determines the period and phase shift? I know that 4=Amplitude The answer in the back of the textbook says Period=2pie/3 Phase Shift=-pie/12
  12. A

    What is the origin of the term sine?

    Origin of the term "sine" It is well-known that "sine" comes from the Latin word "sinus", meaning a "fold", or a pocket. However, its reference to the length of the half-chord on the unit circle remains still rather obscure. I recently came over an explanation that makes perfect sense...
  13. L

    How to determine the point of intersection of sine and cosine?

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  14. R

    Proving slope m of a secant connecting two points of the sine curve

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  15. P

    How do you write equations in a single sine function?

    What would the Domain, Range, Amplitude, Period, Phase shift be for y=-3sin(x)+2cos(x)? How do you write it as a single sine function equivalent to it? :confused:
  16. T

    What is the formula for rise/fall time of a sine wave?

    Does anyone remember the formula for the rise/fall time for a sine wave...? I thought I could calculate it but I did it wrong apparently t1. V_o sin(2*\pi*f*t)=.1 V_o \frac{arcsin(.1)}{2 \pi f} t2. V_o sin(2*\pi*f*t)=.9 V_o \frac{arcsin(.9)}{2 \pi f} t2-t1... but that isn't right
  17. S

    What Are the Characteristics of This Sine Curve Equation?

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  18. X

    How is the RMS of a Sine Wave Derived?

    rms of sine wave = peak * 1/SQRT(2) how is this derived from the rms equation? (Search engines have returned no useful results.)
  19. M

    Struggling with Integrating Sine and Cosine?

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  20. C

    Limit of a sine function problem.

    Part a)I need help understanding a step in solving \lim_{x\rightarrow \infty}xsin(\frac{1}{x}) The textbook is suggesting I replace \frac{1}{x} with y, so that I can get a limit in the form \lim_{y\rightarrow 0}\frac{siny}{y} which is understandably easy to solve. The part I don't...
  21. M

    Evaluate integral of Sine function

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  22. J

    Power of a Sine Wave: Formula to Calculate W

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  23. X

    Relationship between sine and tangent

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  24. P

    Constructing Sine & Cosine Series for f(t)=t

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  25. T

    Horner's scheme for sine Tailor's series computatioin stability

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  26. N

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  27. J

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  28. J

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  29. H

    Why Sine is an odd function and Cosine is an even function?

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  30. R

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  31. B

    S=integral sine S(lnz)^2 integration

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  32. M

    Rules governing the creation of complex waveforms through addition of sine waves

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  33. C

    Sine Waves and Primes: Exploring the Connection Through Music

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  34. A

    Exploring the Sine Function on the Number Plane

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  35. S

    Trig sine substitution doesnt work

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  36. W

    Proving Sine Formula in Triangle ABC

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  37. Link

    Sine function results in singular matrix?

    I am trying to do a sine curve regression on my collected data using my calculator, but when i do it i get an output that says: "Error:Singular matrix". What is happening?
  38. L

    Constructing a Circuit to Limit Voltage of 20V Peak-to-Peak Sine Wave

    How do I construct a circuit that will limit the positive half of a 20V peak-to-peak sine wave to 5.6V and the negative half to -2.5V? So far, I have a 1 k ohm resistor and the diode is forward biased leading to VL and Vout.
  39. T

    Discovering the Cosine Series of Sine: Insights and Calculations Explained

    I tried to find the cosine series of the function f(x) = \sin x, using the equation below: S(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx) where: a_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \cos(nx) dx I found: a_0 = \frac{4}{\pi} a_n = \frac{2 }{\pi (1 - n^2)} (\cos(n \pi) + 1)...
  40. T

    Sine, cosine and tangency of angle 11pi/12

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  41. S

    Defining Sine Waves for Scientists

    Dear All, I have an apparatus that produces a graph of results that looks to me like a sine wave (a half wave plate affecting the power throughput of a laser; it goes from maximum to minimum 4x throughout its 360º rotation). My problem is that I need to define this line as a formula but have...
  42. P

    Understanding Sine, Cosine and Tangent Angles

    Hi people, what is meant by sine, cosine, tangent angles, apart from their general definition of opp side/ hypo side; adj side/ hyp side and so on? how are these sine, cosines and tangent angles invented by humans?
  43. S

    Does the Absolute Value of the Sine Integral Diverge?

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  44. W

    What is the Best Way to Determine x Given y in a Sine Wave Algorithm?

    Hi, This maths stuff is tstarting to hurt my head! :p Ok, I want to use a sine wave to make objects appear at an increasing rate and then a decreasing rate. e.g. where: y=sin(x) y = interval before next object appears so in Maple that'd be: plot(sin(x)+1,x=Pi/2..Pi+(Pi/2)); Now I...
  45. A

    Sine and cosine law in oblique triangles

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  46. R

    Wondering exactly what sine is.

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  47. R

    Time difference between two sine waves

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  48. F

    Sine Wave or Spiral: Exploring the 3-Dimensional Nature of Electromagnetic Waves

    Perhaps this has already been brought up, but is an electromagnetic wave 2-dimensional or 3-dimensional? The current "up-and-down" concept does not have me convinced. If you look at a standard spring, you can see that it is a spiral/helix. You can also see that the side of the spring...
  49. N

    Sine oscillator with unipolar operational amplifier

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  50. S

    Simplifying with Cosine and Sine

    Simplify in terms of cosine and sine only. tan^2x - {csc^2x\over cot^2x} From here, I assume you can flip the fraction and make it {tan^2x\over sin^2x} next, i reduce it to: (sec^2x-1) - ({sec^2x -1\over 1 - cos^2x}) and anyway..im lost; i don't know where to stop and how...
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