Cosine

The cosine function is one of the basic functions encountered in trigonometry (the others being the cosecant, cotangent, secant, sine, and tangent). Let be an angle measured counterclockwise from the x-axis along the arc of the unit circle. Then is the horizontal coordinate of the arc endpoint.

The common schoolbook definition of the cosine of an angle 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.,

(1)

A convenient mnemonic for remembering the definition of the sine, cosine, and tangent is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent).

As a result of its definition, the cosine function is periodic with period . By the Pythagorean theorem, also obeys the identity

(2)

Min Max Re Im

The definition of the cosine function can be extended to complex arguments using the definition

(3)

where e is the base of the natural logarithm and i is the imaginary number. Cosine is an entire function and is implemented in the Wolfram Language as Cos[z].

A related function known as the hyperbolic cosine is similarly defined,

(4)

The cosine function has a fixed point at 0.739085... (OEIS A003957), a value sometimes known as the Dottie number (Kaplan 2007).

The cosine function can be defined analytically using the infinite sum

			(5)
			(6)

or the infinite product

(7)

A close approximation to for is

			(8)
			(9)

(Hardy 1959), where the difference between and Hardy's approximation is plotted above.

The cosine obeys the identity

(10)

and the multiple-angle formula

(11)

where is a binomial coefficient. It is related to via

(12)

(Trott 2006, p. 39).

Summation of from to can be done in closed form as

			(13)
			(14)
			(15)
			(16)
			(17)

Similarly,

(18)

where . The exponential sum formula gives

			(19)
			(20)

The sum of can also be done in closed form,

(21)

The Fourier transform of is given by

			(22)
			(23)

where is the delta function.

Cvijović and Klinowski (1995) note that the following series

(24)

has closed form for ,

(25)

where is an Euler polynomial.

A definite integral involving is given by

(26)

for where is the gamma function (T. Drane, pers. comm., Apr. 21, 2006).

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