Class 12 Maths Inverse Trigonometric Functions Inverse of Cosec function

Inverse of Cosec function

• Natural domain & Range of cosec function, cosec : R – { x : x = nπ, n Z} → R – (– 1, 1)
• If we restrict domain to [-π/2 , π/2 ] – {0}, then it becomes one-one & onto.
• Restricted domain & range of cosec function, cosec: [-π/2 , π/2 ] – {0} → R – (– 1, 1)
• Restricted domain & range of cosec-1 function, cosec -1 : R – (– 1, 1) à [-π/2 , π/2 ] – {0}
• 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
• If y = cosec–1 x, then cosec y = x.
• 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 .