Domain restrictions are necessary to create inverse trigonometric functions rather than relations because inverse trigonometric functions are functions themselves, and functions by definition must map each element in the domain to exactly one element in the codomain. Without domain restrictions, the inverse trigonometric functions would not be well-defined, as there could be multiple values in the domain that map to the same value in the codomain.
For example, in the case of the sine function, if we did not have domain restrictions, we could have multiple angles that have the same sine value (such as 30 degrees and 150 degrees both having a sine of 0.5). This would make it impossible to define a unique inverse function that maps each sine value back to a unique angle.
By imposing domain restrictions, such as restricting the domain of the sine function to a specific interval (e.g. [-π/2, π/2] for arcsin), we ensure that each element in the domain maps to exactly one element in the codomain, allowing us to define unique inverse trigonometric functions.
Explain why domain restrictions are necessary to create inverse trigonometric functions rather than relations
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