User:Qzekrom/Cosine


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In mathematics, the cosine function is a function of an angle. In a unit circle, cosine gives the distance of a point on said circle from the y-axis, while the sine function gives its distance from the x-axis. The function can be equivalently defined as the ratio of the length of the leg adjacent to an angle of a right triangle to that of the hypotenuse.

Cosine is commonly listed after sine, second amongst the trigonometric functions.

Like the sine function, the cosine function is used to model periodic phenomena such as waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and temperature variations throughout the year.

The cosine function is the derivative of the sine function. This can be proved using differentiation by parts:

$$ \begin{align} \cos x = \frac{d}{dx}\sin x = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} \\ = \frac{d}{dx}x - \frac{d}{dx} \frac{x^3}{3!} + \frac{d}{dx} \frac{x^5}{5!} - \frac{d}{dx} \frac{x^7}{7!} + \cdots \\ = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\ = \sum_{k = 0}^\infty \frac{(-1)^n}{(2n)!}x^{2n} \end{align} $$

This can also be proven using geometry and limits.