Introduction:
We know that not every function as an inverse. A function has an inverse if and only if it is one to one and onto. As all trigonometric functions are periodic, they are all many to one type of functions. So technically, inverse of trig functions do not exist. But if we can suitably restrict the domain of the trigonometric function , then it becomes one to one and onto. Therefore with this modified domain the trigonometric function can have an inverse. Let us look at the following examples:
Inverse trig functions examples:
1. Inverse of sine function:
The sine function defined as follows: sin = {(x,y) | y = sin(x), x belongs to R, y belongs to [-1,1]} is an onto function. It is a many to one function. It is a periodic function with at period of 2belongs tobelongs to. But instead of R, if the domain is restricted to say, [-pi/2,pi/2], or [pi/2,3*pi/2], or [3pi/2, 5pi/2] etc, then it becomes one to one and still remains onto. Thus now we can define the inverse of sin function using any one of the above domains as follows: sin^(-1) = [(y,x) | y = sin(x), x belongs to [-pi/2,pi/2], y belongs to [-1,1]}, where “sin^(-1)” is the symbol for inverse sine function.
2. Inverse of cosine function:
Just like how we defined the inverse of sine function, we can define the inverse of cosine function by restricting its domain as well. The domain restrictions can be made to suite our purpose. Therefore, inverse cosine function can be defined as follows: cos^(-1) = [(y,x) | y = cos(x), x belongs to [0,pi], y belongs to [-1,1]}
The other trigonometric function inverses can also be defined similarly.
Interrelations between inverse trigonometric functions:
Sin^(-1) x = cosec^(-1)(1/x), or cosec^(-1)(x) = sin^(-1)(1/x). The inverse functions of cos and sec and the inverse function of tan and cot are related the same way.
Integral of inverse trig functions:
To find integral of inverse trigonometric functions, we use the method of integration by parts. We know that we follow the order LIATE (Logarithmic, Inverse Trigonometric, Algebraic, Trigonometric, Exponential function) for integration by parts. When we integrate inverse trigonometric functions, the inverse function becomes the u of the integration by parts and since there is no other function it is multiplied to, we take v = 1. Thus for example, if we were to integrate the function like tan^(-1)x using the integration by parts rule, then here u = tan^(-1)x and v = 1 and then integrate using integration by parts.
