Angle Sine Function

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A Angle Sine Function is a trigonometric function and a periodic function with period [math]2\pi[/math]

  • Context:
    • It can be defined as the ratio between lengths of the opposite side ([math]a[/math]) to the acute angle [math] \theta[/math] and the hypotenuse ([math]h[/math]) in a right triangle, [math]\sin(\theta)= \frac{a}{h}[/math].
    • It can be defined as the imaginary part of the complex exponential function [math] \sin (\theta) = Im\left[e^{i\theta}\right] [/math]
    • It can also be defined as by the following power series, for any real number ([math]x[/math]), [math] \sin(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots [/math].
    • It can also be represented as a generalized continued fraction [math] \sin (x) = \cfrac{x}{1 + \cfrac{x^2}{2\cdot3-x^2 + \cfrac{2\cdot3 x^2}{4\cdot5-x^2 + \cfrac{4\cdot5 x^2}{6\cdot7-x^2 + \ddots}}}}. [/math].
    • It must satisfy the following properties, where [math]cos(x)[/math] is the cosine function, [math]\binom nk[/math] is the binomial coefficient, [math]\Gamma(x)[/math] is the gamma function, C is a constant, x and y are real numbers:
      1. [math]\sin^2 (x) + \cos^2 (x) = 1 \quad[/math] Pythagorean identity
      2. [math]\sin(\theta)=\sin(\theta+ 2\pi k)\quad[/math] Periodic Function
      3. [math]\sin\left(x+y\right)=\sin x \cos y + \cos x \sin y\quad[/math] Sum
      4. [math]\sin\left(x-y\right)=\sin x \cos y - \cos x \sin \quad[/math] Difference
      5. [math]\sin\left(2x\right)= 2 \sin x \cos x\quad[/math] Double-angle formula
      6. [math]\sin\left(nx\right)= \sum_{k=0}^n\binom nk \cos^k x\;\sin^{n-k} x\;\sin(\frac{1}{2}(n-k)\pi) \quad[/math] multiple-angle formula
      7. [math]\frac{d}{dx}\sin(x) = \cos(x)\quad[/math] Derivative
      8. [math]\int\sin(x)\;\mathrm{d}x = -\cos(x)+C\quad [/math] Indefinitive integral
      9. [math]\int_0^\infty \sin(x^n)\;\mathrm{d}x = \Gamma(1+\frac{1}{n})sin\left(\frac{\pi}{2n}\right)\quad [/math] Definitive integral
  • Example(s):
    • [math]\sin(\theta) = \cos\left(\pi/2 - \theta \right)[/math] , where [math]cos(x)[/math] is the cosine function
    • [math]\sin(\theta) = \pm\sqrt{1 - \cos^2(\theta)}[/math]
    • [math]\sin(\theta) = 1 / \csc(\theta) [/math], where [math]\csc(x)[/math] is the cosecant function
    • [math] \sin(\theta) = (e^{i\theta}-e^{-i\theta})/2i=\sinh(i\theta)/i[/math], where [math]i[/math] is the imaginary number and [math]sinh(x)[/math] is the hyperbolic sine function
    • [math]\sin(\theta)= \pm\frac{1}{\sqrt{1 + \cot^2(\theta)}}[/math], where [math]\cot(x)[/math] is the cotangent function
    • [math]\sin(\theta) = \pm\frac{\tan(\theta)}{\sqrt{1 + \tan^2(\theta)}} [/math] , where [math]tan(x)[/math] is the tangent function
    • [math]\sin(\theta)= \pm\frac{\sqrt{\sec^2(\theta) - 1}}{\sec(\theta)} [/math] , where [math]sec(x)[/math] is the secant function
  • Counter-Example(s):
  • See: Cosine Function, Cosecant Function, Cotangent Function, Secant Function, Tangent Function, Complex Exponential Function, Hyperbolic Sine Function, Pythagorean Theorem, Pythagorean identity


References

2015

1999

  • (Wolfram Mathworld , 1999) ⇒ http://mathworld.wolfram.com/Sine.html
    • QUOTE: The sine function [math]sin\;x[/math] is one of the basic functions encountered in trigonometry (the others being the cosecant, cosine, cotangent, secant, and tangent). Let [math]\theta[/math] be an angle measured counterclockwise from the x-axis along an arc of the unit circle. Then [math]sin\;\theta[/math] is the vertical coordinate of the arc endpoint, as illustrated in the left figure above [1].

      The common schoolbook definition of the sine of an angle theta in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths of the side of the triangle opposite the angle and the hypotenuse, i.e., :[math]sin\;\theta=opposite/hypotenuse[/math]