cosine

cosine, one of the six trigonometric functions, which, in a right triangle ABC, for an angle A, is cos A = length of side adjacent to angle A/length of hypotenuse.(The other five trigonometric functions are sine [sin], tangent [tan], secant [sec], cosecant [csc], and cotangent [cot].)

From the definition of the sine of angle A, sin A = length of side opposite to angle A/length of hypotenuse,and the Pythagorean theorem, one has the useful identity sin² A + cos² A = 1.Other useful identities involving the cosine are the half-angle formula, cos (A/2) = 1 + cos A/2;and the double-angle formula,cos 2A = cos² A − sin² A.

The law of cosines is a generalization of the Pythagorean theorem relating the lengths of the sides of any triangle. If a, b, and c are the lengths of the sides and C is the angle opposite side c, then c² = a² + b² − 2ab cos C. The reciprocal of the cosine is the secant: 1/cos A = sec A.

The cosine function has several other definitions. If a circle with radius 1 has its centre at the origin (0,0) and a line is drawn through the origin with an angle A with respect to the x-axis, the sine is the x-coordinate of the point where the line intersects the circle. When A is expressed in radians, the cosine function has a period of 2π. The function has a maximum value of 1 at 0 and a minimum of −1 at π; it has a value of 0 at π/2 and 3π/2. Also, cos (−A) = cos A.

The cosine can also be expressed as the power seriescos x = 1 − x²/2! + x⁴/4! − x⁶/6! + ⋯,where the exclamation point indicates the factorial function. When combined with a similar power function for the sine function, one obtains Euler’s identity, e^ix = cos x + i sin x, where e is the base of the natural logarithm and i is the square root of −1. When x is equal to π or 2π, the formula yields two elegant expressions relating π, e, and i: e^iπ = −1 and e^2iπ = 1, respectively.

With respect to x, the derivative of cos x is −sin x, and the indefinite integral of sin x is sin x.