Inverse of Sec inverse
o Cosec function restricted to any of intervals [ −3π / 2 , -π / 2 ] – { -π ,} to [ π / 2 , 3π / 2 ] - { π } , is one-one & its range is [– 1 , 1 ]. Corresponding to each such interval , we get a branch of function cosec – 1 . The branch with range , [ -π / 2 , π / 2 ] – { 0 } is called principal value branch
o If y = cosec – 1 x , then cosec y = x .
o
Thus , the graph of cosec – 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 cosec function , then ( b , a ) becomes the corresponding point on the graph of inverse of cosec function
Inverse of Sec inverse
o Natural domain & Range of sec function : R – { x : x = ( 2n + 1 ) π / 2 , n ∈ Z } → R – (– 1 , 1 ) o
If we restrict domain to [ 0 , π ] –{ π / 2 }, then it becomes one-one & onto with range R – (– 1 , 1 ).
o Restricted domain & range of sec function , sec : [ 0 , π ] –{ π / 2 } → R – (– 1 , 1 ) o Restricted domain & range of sec -1 function , sec -1 : R – (– 1 , 1 ) à [ 0 , π ] –{ π / 2 }
o Actually , sec function restricted to any of the intervals [– π , 0 ] - { - π / 2 } , [ π , 2π ] - { 3π / 2 } , is one-one & its range is R – (– 1 , 1 ). Corresponding to each such interval , we get a branch of function sec – 1 . The branch with range , [ 0 , π ], is called principal value branch
o If y = sec – 1 x , then sec y = x .