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 ]
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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.
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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
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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 }