Angle Cosine Function

An Angle Cosine 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 adjacent side ([math]\displaystyle{ b }[/math]) to the acute angle [math]\displaystyle{ \theta }[/math] and the hypotenuse ([math]\displaystyle{ h }[/math]) in a right triangle, [math]\displaystyle{ \cos(\theta)= \frac{b}{h} }[/math].
  • It can be defined as the real part of the complex exponential function [math]\displaystyle{ \cos (\theta) = Re\left[e^{i\theta}\right] }[/math]
  • It can be defined as by the following power series, for any real number ([math]\displaystyle{ x }[/math]) [math]\displaystyle{ \cos(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots }[/math].
  • It must satisfy the following properties, where [math]\displaystyle{ sin(x) }[/math] is the sine 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{ \cos(\theta)=\cos(\theta+ 2\pi k)\quad }[/math] Periodic Function
    3. [math]\displaystyle{ \cos\left(x+y\right)=\cos x \cos y - \sin x \sin y\quad }[/math] Sum
    4. [math]\displaystyle{ \cos\left(x-y\right)=\cos x \cos y + \sin x \sin y \quad }[/math] Difference
    5. [math]\displaystyle{ \cos\left(nx\right)=\sum_{k=0}^n\binom nk \cos^k(x)\sin^{n-k}(x) cos[1/2(n-k)\pi], \quad }[/math] multiple-angle formula
    6. [math]\displaystyle{ \frac{d}{dx}\cos(x) = - \sin(x)\quad }[/math] Derivative
    7. [math]\displaystyle{ \int\cos(x)\;\mathrm{d}x = \sin(x)+C\quad }[/math] Indefinitive integral
    8. [math]\displaystyle{ \int_0^\infty \cos(x^n)\;\mathrm{d}x = \Gamma(1+\frac{1}{n})\cos\left(\frac{\pi}{2n}\right)\quad }[/math] Definitive integral
• Example(s):
  • [math]\displaystyle{ \cos(\theta) = \sin\left(\pi/2 + \theta \right) }[/math] , where [math]\displaystyle{ sin(x) }[/math] is the sine function.
  • [math]\displaystyle{ \cos(\theta) = \pm\sqrt{1 - \sin^2(\theta)} }[/math]
  • [math]\displaystyle{ \cos(\theta) = 1 / \sec(\theta) }[/math], where [math]\displaystyle{ sec(x) }[/math] is the secant function.
  • [math]\displaystyle{ \cos(\theta) = (e^{i\theta}+e^{-i\theta})/2=\cosh(i\theta) }[/math], where [math]\displaystyle{ i }[/math] is the imaginary number and [math]\displaystyle{ sinh(x) }[/math] is the hyperbolic cosine function.
  • [math]\displaystyle{ \cos(\theta)= \pm\frac{\cot (\theta)}{\sqrt{1 + \cot^2 (\theta)}} }[/math], where [math]\displaystyle{ \cot(x) }[/math] is the cotangent function.
  • [math]\displaystyle{ \cos(\theta) = \pm\frac{1}{\sqrt{1 + \tan^2 (\theta)}} }[/math] , where [math]\displaystyle{ tan(x) }[/math] is the tangent function.
  • [math]\displaystyle{ \cos(\theta)= \pm\frac{\sqrt{\csc^2(\theta) - 1}}{\csc(\theta)} }[/math] where [math]\displaystyle{ csc(x) }[/math] is the cosecant function.
• Counter-Example(s):
  • Sine Function.
  • Cosecant Function.
  • Cotangent Function.
  • Secant Function.
  • Tangent Function.
• See: Sine Function, Cosecant Function, Cotangent Function, Secant Function, Tangent Function, Complex Exponential Function, Hyperbolic Cosine Function, Pythagorean Theorem, Pythagorean identity, Angle, Dot Product, Distance Function.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Cosine_(trigonometric_function)
  • QUOTE: The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case
    • [math]\displaystyle{ \cos A = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} = \frac {b} {h}\, }[/math].

1999

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

The common schoolbook definition of the cosine 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 adjacent to the angle and the hypotenuse, i.e.,

[math]\displaystyle{ cos(\theta)=adjacent/hypotenuse }[/math].

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