 Trigonometric (cyclometric) functions and inverse trigonometric functions (arc functions)
The graphs of the trigonometric functions and inverse trigonometric functions or arc-functions
The graph of the tangent function and the cotangent function
The graph of the arc-tangent function and the arc-cotangent function
Trigonometric (cyclometric) functions and inverse trigonometric functions (arc functions)
Trigonometric functions are defined as the ratios of the sides of a right triangle containing the angle equal to the argument of the function in radians.
Or more generally for real arguments, trigonometric functions are defined in terms of the coordinates of the terminal point Q of the arc (or angle) of the unit circle with the initial point at P(1, 0).        sin2x + cos2x = 1
-  The tangent function  y = tan x is the ratio of the y-coordinate to the x-coordinate of the terminal point of the arc x of the unit circle, or it is the ratio of the sine function to the cosine function.
In a right-angled triangle the tangent function is equal to the ratio of the length of the side opposite the given angle to that of the adjacent side.
-  The cotangent function  y = cot x is the reciprocal of the tangent function, or it is the ratio of the cosine function to the sine function.
In a right-angled triangle the cotangent function is equal to the ratio of the length of the side adjacent to the given angle to that of the side opposite it.
The graph of the tangent function and the cotangent function    -  The arc-tangent function  y = tan-1x or  y = arctan x is the inverse of the tangent function, so that its value for any argument is an arc (angle) whose tangent equals the given argument.
 That is,  y = tan-1x if and only if  x = tan y.  For example, Thus, the arc-tangent function is defined for all real arguments, and its principal values are by convention taken to be those strictly between -p/2 and p/2.
-  The arc-cotangent function  y = cot-1x or  y = arccot x is the inverse of the cotangent function, so that its value for any argument is an arc (angle) whose cotangent equals the given argument.
 That is,  y = cot-1x if and only if  x = cot y.  For example, Thus, the arc-cotangent function is defined for all real arguments, and its principal values are by convention taken to be those strictly between 0 and p.
The graph of the arc-tangent function and the arc-cotangent function     Intermediate algebra contents 