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.