Trigonometric Functions

The trigonometric functions, also called circular functions, are the sine, cosine, tangent, cotangent, secant, and cosecant. For a real angle , the sine and cosine give the vertical and horizontal Cartesian coordinates, respectively, of the corresponding point on the unit circle. The remaining functions are obtained from these by quotients and reciprocals, wherever the displayed expressions are defined:

			(1)
			(2)
			(3)
			(4)
			(5)

The figure above combines the six standard functions in one unit circle construction. Let and , where . The drawing uses . In this construction, the tangent line at meets the horizontal and vertical axes at and , respectively. The radius has unit length, and the labeled line segments have the corresponding positive lengths. The figure also shows the versine , coversine , exsecant , and excosecant .

The following table pairs unit circle constructions with function graphs. For the constructions, let , , , and , where . Let be the orthogonal projection of onto . The ray meets the tangent lines at and in and , respectively. The blue line segment in each construction has the indicated length. The drawings use , with orange points marking the same angle on the unit circle and function graph.

unit-circle construction	graph	function	segment length	period	range
		sine
		cosine
		tangent			all real
		cotangent			all real
		secant
		cosecant

The function graphs use radians. The periods are the least positive ones, and the ranges are taken over the real numbers where each function is defined. The tangent and secant are undefined at , while the cotangent and cosecant are undefined at , where is an integer.

The hyperbolic functions are closely related analogs obtained by removing factors of from the corresponding formulas in terms of . Two distinct generalizations associated with unit superellipses are the p-trigonometric functions and the area-parameterized squigonometric functions. Other notations are sometimes used, as summarized in the following table.

	alternate notations
	(Erdélyi et al. 1981, p. 7), (Gradshteyn and Ryzhik 2000, p. xxix)
	(Gradshteyn and Ryzhik 2000, p. xxvii)
	(Gradshteyn and Ryzhik 2000, p. xxix)

The inverses of these functions (the inverse trigonometric functions) are denoted , , , , , and . Note that the notation here means inverse function, not to the power.