Inverse Trigonometric Functions are, like any other inverse function, mathematical operators that undo another function's operation.
Given a triangle like this
the basic trigonometric functions would be defined as:
with the angle as their input (or argument) and a ratio of sides as their result. However, the inverse functions take the ratio as input and return the angle:
This means the inverse trigonometric functions are useful whenever we know the sides of a triangle and want to find its angles.
Note: The notation might be confusing, as we normally use a negative exponent to indicate the reciprocal. However, in this case, . When we want the reciprocal of we use . In order to avoid this ambiguity, sometimes people might choose to write the inverse functions with an arc prefix. For example:
In following statement, and are positive, co-prime integers. What is the sum of and ?
Since we are trying to find and , we should take the tangent of both sides of the equation:
Further, we we can use the ratio given to sketch the triangle with in it, using the definition of :
Now, using the Pythagorean theorem, we can see that . This means . Finally, we evaluate , which means .