Explain why domain restrictions are necessary to create inverse trigonometric functions rather than relations.
Domain restrictions are necessary to create inverse trigonometric functions rather than relations because inverse functions must satisfy specific criteria to be considered functions. Inverse functions are functions that undo the action of the original function, and therefore must have a one-to-one correspondence with the original function.
Trigonometric functions such as sine, cosine, and tangent are not one-to-one functions because they have multiple outputs for the same input (i.e., they are not injective). Therefore, in order to create inverse trigonometric functions, we must restrict the domain of the original trigonometric functions to make them one-to-one.
For example, the sine function has a period of 2π, which means that it repeats its values every 2π units. To create an inverse function for sine, we must restrict the domain of sine to a specific interval (e.g., [-π/2, π/2]) where it is one-to-one. This allows us to define the inverse sine function, also known as arcsin, as a function that maps each output of sine to a unique input.
By imposing these domain restrictions, we can create inverse trigonometric functions that satisfy the criteria for functions and have a well-defined inverse relationship with the original trigonometric functions.