cos1x

cos1x is a notation sometimes encountered in elementary trigonometry and signal processing, representing the cosine function applied to the product of the constant 1 and the variable x. Because multiplication by one does not alter the value of the argument, cos1x is equivalent to the standard cosine function cos x. The expression appears in textbooks or software that automatically prepend a coefficient to the variable, and it serves to emphasize the linear relationship between the angle and its measurement unit.

The cosine function is defined for real numbers x as the x‑coordinate of a point on the unit circle that subtends an angle x measured in radians from the positive x‑axis. Its values range between –1 and 1, and it is an even, periodic function with period 2π, satisfying cos(–x) = cos x and cos(x + 2π) = cos x for all real x. The function obeys the fundamental trigonometric identity cos²x + sin²x = 1 and can be expressed as a power series: cos x = Σ_{k=0}^∞ (–1)^k x^{2k} / (2k)!.

In applied contexts, cos1x may appear in the analysis of oscillatory phenomena such as alternating currents, sound waves, and mechanical vibrations, where the argument x often represents time or angular displacement. Because the coefficient of the variable is 1, the angular frequency of the resulting sinusoid is one radian per unit of the independent variable, making the period exactly 2π units. Consequently, any properties derived for cos x apply directly to cos1x without modification.