Angle Sine Function

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

• Context:
  • It can be defined as the ratio between lengths of the opposite side ([math]\displaystyle{ a }[/math]) to the acute angle [math]\displaystyle{ \theta }[/math] and the hypotenuse ([math]\displaystyle{ h }[/math]) in a right triangle, [math]\displaystyle{ \sin(\theta)= \frac{a}{h} }[/math].
  • It can be defined as the imaginary part of the complex exponential function [math]\displaystyle{ \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]\displaystyle{ x }[/math]), [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ cos(x) }[/math] is the cosine function, [math]\displaystyle{ \binom nk }[/math] is the binomial coefficient, [math]\displaystyle{ \Gamma(x) }[/math] is the gamma function, C is a constant, x and y are real numbers:
    1. [math]\displaystyle{ \sin^2 (x) + \cos^2 (x) = 1 \quad }[/math] Pythagorean identity
    2. [math]\displaystyle{ \sin(\theta)=\sin(\theta+ 2\pi k)\quad }[/math] Periodic Function
    3. [math]\displaystyle{ \sin\left(x+y\right)=\sin x \cos y + \cos x \sin y\quad }[/math] Sum
    4. [math]\displaystyle{ \sin\left(x-y\right)=\sin x \cos y - \cos x \sin \quad }[/math] Difference
    5. [math]\displaystyle{ \sin\left(2x\right)= 2 \sin x \cos x\quad }[/math] Double-angle formula
    6. [math]\displaystyle{ \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]\displaystyle{ \frac{d}{dx}\sin(x) = \cos(x)\quad }[/math] Derivative
    8. [math]\displaystyle{ \int\sin(x)\;\mathrm{d}x = -\cos(x)+C\quad }[/math] Indefinitive integral
    9. [math]\displaystyle{ \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]\displaystyle{ \sin(\theta) = \cos\left(\pi/2 - \theta \right) }[/math] , where [math]\displaystyle{ cos(x) }[/math] is the cosine function.
  • [math]\displaystyle{ \sin(\theta) = \pm\sqrt{1 - \cos^2(\theta)} }[/math]
  • [math]\displaystyle{ \sin(\theta) = 1 / \csc(\theta) }[/math], where [math]\displaystyle{ \csc(x) }[/math] is the cosecant function.
  • [math]\displaystyle{ \sin(\theta) = (e^{i\theta}-e^{-i\theta})/2i=\sinh(i\theta)/i }[/math], where [math]\displaystyle{ i }[/math] is the imaginary number and [math]\displaystyle{ sinh(x) }[/math] is the hyperbolic sine function.
  • [math]\displaystyle{ \sin(\theta)= \pm\frac{1}{\sqrt{1 + \cot^2(\theta)}} }[/math], where [math]\displaystyle{ \cot(x) }[/math] is the cotangent function.
  • [math]\displaystyle{ \sin(\theta) = \pm\frac{\tan(\theta)}{\sqrt{1 + \tan^2(\theta)}} }[/math] , where [math]\displaystyle{ tan(x) }[/math] is the tangent function.
  • [math]\displaystyle{ \sin(\theta)= \pm\frac{\sqrt{\sec^2(\theta) - 1}}{\sec(\theta)} }[/math] , where [math]\displaystyle{ sec(x) }[/math] is the secant function.
• Counter-Example(s):
  • Cosine Function.
  • Cosecant Function.
  • Cotangent Function.
  • Secant Function.
  • Tangent Function.
• See: Cosine Function, Cosecant Function, Cotangent Function, Secant Function, Tangent Function, Complex Exponential Function, Hyperbolic Sine Function, Pythagorean Theorem, Pythagorean identity.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Sine
  • QUOTE: Sine, in mathematics, is a trigonometric function of an angle. The sine of an 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 (i.e., the hypotenuse).

1999

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

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