cosine

Cosine, denoted cos, is a fundamental trigonometric function. In a unit circle, cos θ is the x-coordinate of the point obtained by rotating (1,0) by angle θ, and in a right triangle it is the ratio of the length of the adjacent side to the hypotenuse. It is an even function with period 2π and range [−1,1]. Its graph is a smooth periodic wave.

Key identities include sin^2 θ + cos^2 θ = 1, cos(−θ) = cos θ, and the angle addition formula cos(a+b) = cos a cos b − sin a sin b. The double-angle identity cos 2x = 2 cos^2 x − 1 (equivalently 1 − 2 sin^2 x) is widely used. The powers of cos can be rewritten with the half-angle identities cos^2 x = (1+cos 2x)/2 and sin^2 x = (1−cos 2x)/2.

Calculus and analysis properties: the derivative of cos x is −sin x, and the second derivative is −cos x, making cos a solution to the simple harmonic differential equation y'' + y = 0. The integral ∫ cos x dx = sin x + C. The Maclaurin series for cosine is cos x = ∑_{n=0}^∞ (−1)^n x^{2n}/(2n)!.

In vector terms, cos θ equals the dot product of two unit vectors forming angle θ: cos θ = (u·v)/(||u|| ||v||). It also represents the projection of one vector onto another and appears in physics to describe angular relationships and oscillatory motion. Cosine is a core component of Fourier analysis and many geometric and physical applications.