Cosine

Cosine is a trigonometric ratio. In a right triangle with an angle $\theta$ ,

$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$

$\text{Adjacent}$ is the side of the triangle that is part of angle $\theta$ , and $\text{hypotenuse}$ is the side opposite the right angle.

Properties

The cosine of an angle is the x-coordinate of the point of intersection of said angle and a unit circle.

As a result of Euler's formula, the cosine function can also be represented as

$\cos(\theta)=\frac{e^{i \theta}+e^{-i \theta}}{2}$

The reciprocal of cosine is secant (abbreviated as $\sec$), while its inverse is $\arccos$ or $\cos^{-1}$ . Note that sine is not being raised to the power of -1; this is an inverse function, not a reciprocal.

The derivative of $\cos(x)$ is $-\sin(x)$ , while its antiderivative is $\sin(x)$ .

The trigonometric addition/subtraction identiy of cosine is:

$\cos(a\pm b)=\cos(a)\cos(b)\mp\sin(a)\sin(b)$

The square of cosine:

$\cos^2(\theta)=\frac{\cos(2\theta)+1}{2}$

Imaginary number:

$\cos(ix)=\cosh(x)$

Cosine is an even function:

$cos(x)=cos(-x)$

Limits

$\lim_{x\to 0} \frac{1 - cos x}{x} = 0$

Power series

More formally defined, the cosine is defined by the power series, which is convergent for all $x$,

$\cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}$

Where $n!$ denotes the factorial of n. From this, the above properties may be derived.

History

The cosine function was first discovered in medieval Islamic mathematics. In the early 9th century, Persian mathematician Muhammad ibn Mūsā al-Khwārizmī (c. 780-850) produced a table for the cosine function.

Something equivalent to the spherical law of cosines was used (but not stated in general) by Al-Khwārizmī (9th century) and Al-Battānī (9th century). The law of cosines was also used (but not stated in general) for the solution of triangles, in the course of solving astronomical problems by al-Bīrūnī (11th century).

The 13th century Persian mathematician Naṣīr al-Dīn al-Ṭūsī, in his Kitāb al-Shakl al-qattāʴ (Book on the Complete Quadrilateral, c. 1250), gave a method for finding the third side of a general scalene triangle given two sides and the included angle by dropping a perpendicular from the vertex of one of the unknown angles to the opposite base, reducing the problem to solving a right-angled triangle by the Pythagorean theorem. If written out using modern mathematical notation, the resulting relation can be algebraically manipulated into the modern law of cosines.

Jamshīd al-Kāshī (1393-1449) later provided the first explicit statement of the law of cosines in a form suitable for triangulation. As such, the law of cosines is known the théorème d'Al-Kashi in France.

In 1972, Nasir Ahmed invented the discrete cosine transform (DCT), an algorithm that utilizes the cosine function. DCT forms the basis of modern digital media, including digital images, audio and video.

See also

• Hyperbolic cosine
• Law of cosines
• Sine
• Secant
• Tangent