. . The i nverse sine function and its properties The sine function is not one-to-one in the entire domain . This is visualized from the fact that every horizontal line y −≤ intersects the graph of y = sin infinitely many times.
In other words , the sine function does not pass the horizontal line test, which is a tool to decide the one-to-one status of a function. If the domain is restricted to − , then the sine function becomes one to one and onto (bijection) with the range [ , ] - . Now, let us define the inverse sine function with [ , ] - as its domain and with − as its range. Definition .
For −≤ , define sin - x as the unique number y in − such that sin y . In other words, the inverse sine function sin : [ , ] →− π π is defined by sin ( ) x y if and only if sin y and y ∈− Note (i) The sine function is one-to-one on the restricted domain − , but not on any larger interval containing the origin. (ii) The cosine function is non-negative on the interval − , the range of sin - x . This observation is very important for some of the trigonometric substitutions in Integral Calculus.
(iii) Whenever we talk about the inverse sine function, we have, sin :