TopThe inverse of a function is the reverse plotting of the Numbers from The Range set to the Domain Set. Similar is the case with Trigonometric Functions like tan, sin, cos, sec, etc. The graph of tan-1 function is drawn on the x-axis across the same axis.
As the inverse tangent function is a multivalued function, we need a branch cut in the complex plane to resolve its values.
We can mathematically express the tan-1 function as,
tan-1 y = 1 / 2 i [ ln ( 1- i y) – ln ( 1 + i y ) ],here, ‘i’ represents for complex numbers. As we discussed the branch cut provides the range for tan-1 function.
The range for x – axis along positive direction is,
Range = ( - ∏/2 , ∏/2 )
The most common range of tan -1 in other quadrants is in the interval (∏ , 0 ).
There are some common values for the inverse tan function as,
tan-1 ( - i ) = - 1 ∞,
tan-1 ( - i ) = - i ∞,
tan -1 0 = 0,
tan -1 i = i ∞,
The differentiation of the inverse tan function is defined mathematically as:
d ( tan-1 y ) / dy = 1 / ( 1+ y2 ).
Similarly, the Integration is obtained by the reverse process,
∫ tan -1 y dy = y tan -1 y – 1 / 2 lm ( 1 + y2 ) + k,
‘k’ is the integration constant.
In the form of complex numbers,
The inverse tan is,
∅=tan -1 ( y / x ),
Here, ∅is the angle measured in the clockwise direction.