cosine
- Related Topics:
- secant
- trigonometric function
- law of cosines
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.
Did You Know?
Cosine functions help describe periodic motion. They model simple harmonic motion, such as the swing of a pendulum, vibrations of a tuning fork, and the movement of a mass hanging from a spring. Sound waves and alternating current (AC) follow sinusoidal patterns, which can be represented using cosine functions.
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 center at the origin (0,0) and a line is drawn through the origin with an angle A with respect to the x-axis, the cosine 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, eix = cos x + i sin xwhere 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: eiπ = −1 and e2iπ = 1, respectively.
Euler’s formula provides a direct way to express the cosine function in terms of exponential functions:cos x = eix+ e-ix/2
With respect to x, the derivative of cos x is −sin x, and the indefinite integral of cos x is sin x.
