Trigonometry
      Inverse Trigonometric Functions or Arc-functions and their Graphs
         Inverse functions
      The arc-sine function and the arc-cosine function
         The arc-sine function
         The arc-cosine function
         The graph of the arc-sine function and the arc-cosine function
      The arc-tangent function and the arc-cotangent function
         The arc-tangent function
         The arc-cotangent function
         The graph of the arc-tangent function and the arc-cotangent function
      The arc-cosecant function and the arc-secant function
         The graph of the arc-cosecant and the arc-secant function
Inverse Trigonometric Functions or Arc-functions and their Graphs
Inverse functions
The inverse function, usually written f -1, is the function whose domain and the range are respectively the range and domain of a given function f, that is
f -1(x) = y  if and only if   f (y) = x .
Thus, the composition of the inverse function and the given function returns x, which is called the identity function, i.e.,
f -1(ƒ(x)) = x    and    f (f -1(x)) = x.
The inverse of a function undoes the procedure (or function) of the given function.
A pair of inverse functions is in inverse relation.
Example:  If given  f (x) = log2 x  then  f -1(x)  = 2x  since,
   
The arc-sine function and the arc-cosine function
  -  The arc-sine function  y = sin-1x or  y = arcsin x is the inverse of the sine function, so that its value for any argument is an arc (angle) whose sine equals the given argument.
That is,  y = sin-1x if and only if  x = sin y.  For example,
Thus, the arc-sine function is defined for arguments between -1 and 1, and its principal values are by convention taken to be those between -p/2 and p/2.
  -  The arc-cosine function  y = cos-1x or  y = arccos x is the inverse of the cosine function, so that its value for any argument is an arc (angle) whose cosine equals the given argument.
That is,  y = cos-1x if and only if  x = cos y.  For example,
Thus, the arc-cosine function is defined for arguments between -1 and 1, and its principal values are by convention taken to be those between 0 and p.
The graph of the arc-sine function and the arc-cosine function
The arc-tangent function and the arc-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
The arc-cosecant function and the arc-secant function
  -  The arc-cosecant function  y = csc-1x or  y = arccsc x is the inverse of the cosecant function, so that its value for any argument is an arc (angle) whose cosecant equals the given argument.
That is,  y = csc-1x if and only if  x = csc y.  For example,
Thus, the arc-cosecant function is defined for arguments less than -1 or greater than 1, and its principal values are by convention taken to be those between -p/2 and p/2.
  -  The arc-secant function  y = sec-1x or  y = arcsec x is the inverse of the secant function, so that its value for any argument is an arc (angle) whose secant equals the given argument.
That is,  y = sec-1x if and only if  x = sec y.  For example,
Thus, the arc-secant function is defined for arguments less than -1 or greater than 1, and its principal values are by convention taken to be those between 0 and p. 
The graph of the arc-cosecant and the arc-secant function
Pre-calculus contents G
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