Inverse Trigonometry Functions

Also the reciprocal functions of sine and cosine functions have inverses. If the domain of the cosecant function is restricted to the interval R-(-1,1) it becomes 1-1 onto and is invertible with range any one of these intervals of the form [-π/2, π/2]-{0},[-3π/2, -π/2]-{- π },[ π/2,3π/2]-{ π } etc. The branch with range [-π/2, π/2] -{0} is called the principal value branch of cosec-1 function.

In the case of secant function , if the domain is restricted to [0, π ]-{ π/2} then it is one to one onto and its inverse exists. The domain of the sec-1 function is R - (-1,.1) and range is any one of the intervals of the form [-π,0]-{- π/2},[0, π ] -{ π/2},[ π ,2 π ]-{3π/2} etc .The branch with range [0, π]-{ π/2} is called the principal value branch of the sec-1 function.

For the tangent function if the domain is restricted to (-π/2, π/2) it becomes 1-1 onto with range R. Hence tan-1 is a function with domain R and range any one of the intervals of the form (-3 π/2,- π/2),(- π/2, π/2),( π/2,3π/2) etc. The branch with range (-π/2, π/2) is called the principal value branch of tan-1 function. Similarly, if the domain of the cotangent function is restricted to (0, π ) then it becomes 1-1 onto with range R. So cot-1 function is defined with domain R and range any one of the intervals of the form (-π, 0),(0, π ),( π ,2π ) etc. The branch of the function with range (0, π ) is called the principal value branch of the function.

Using trigonometric identities and properties of trigonometric functions we define and derive the properties of inverse trigonometric functions. These properties are used for solving various problems and proving various results in inverse trigonometric functions