Inverse of Cosec function
o If we restrict domain to [ 0 , π ], then it becomes one-one & onto with range [– 1 , 1 ].
o Restricted domain & range of cosine function , cosine : [ 0 , π ] → [– 1 , 1 ] o Restricted domain & range of cos -1 function , cos -1 : [– 1 , 1 ] à [ 0 , π ]
o
Actually , cosine function restricted to any of the intervals [– π , 0 ], [ π , 2π ] , is one-one & its range is [– 1 , 1 ]. Corresponding to each such interval , we get a branch of function cos – 1 . The branch with range , [ 0 , π ], is called principal value branch
o If y = cos – 1 x , then cos y = x .
o
Thus , the graph of cos – 1 function can be obtained from the graph of original function by interchanging x and y axes , i . e ., if ( a , b ) is a point on the graph of cosine function , then ( b , a ) becomes the corresponding point on the graph of inverse of cosine function
Inverse of Cosec function
o
Natural domain & Range of cosec function , cosec : R – { x : x = nπ , n ∈ Z } → R – (– 1 , 1 )
o If we restrict domain to [ -π / 2 , π / 2 ] – { 0 }, then it becomes one-one & onto .
o Restricted domain & range of cosec function , cosec : [ -π / 2 , π / 2 ] – { 0 } → R – (– 1 , 1 )
o Restricted domain & range of cosec -1 function , cosec -1 : R – (– 1 , 1 ) à [ -π / 2 , π / 2 ] – { 0 }