Inverse of Cosine function o Natural domain & Range of cosine function, cosine: R→ [– 1, 1 ]
Natural domain & Range of sine function, sine: R→ [– 1, 1 ]
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If we restrict domain to [ −π / 2, π / 2 ], then it becomes one-one & onto with range [– 1, 1 ].
o Restricted domain & range of sine function, sine: [ −π / 2, π / 2 ] → [– 1, 1 ] o Restricted domain & range of sin-1 function, sine: [– 1, 1 ] à [ −π / 2, π / 2 ]
o Actually, sine function restricted to any of the intervals [ −3π / 2,-π / 2 ], to [ π / 2, 3π / 2 ] etc., is one-one & its range is [– 1, 1 ]. Corresponding to each such interval, we get a branch of function sin – 1. The branch with range, [ −π / 2, π / 2 ], is called principal value branch
o If y = sin – 1 x, then sin y = x.
Graph of an inverse function can be obtained from the corresponding graph of original function as a mirror image( i. e., reflection) along the line y = x.
The graph of sin – 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 sine function, then( b, a) becomes the corresponding point on the graph of inverse of sine function
Inverse of Cosine function o Natural domain & Range of cosine function, cosine: R→ [– 1, 1 ]