Domain restrictions are necessary when creating inverse trigonometric functions from the corresponding trigonometric functions because inverse functions must be unique. Here are the key reasons for these restrictions:
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Many-to-One Nature of Trigonometric Functions: Trigonometric functions like sine, cosine, and tangent are periodic and do not pass the horizontal line test, which means that for any given output (y-value), there can be multiple inputs (x-values). For example, \(\sin(x) = 0.5\) at several angles (e.g., \(30^\circ\) and \(150^\circ\), plus multiples of \(360^\circ\)). To be able to define an inverse function, we need to restrict the input values to ensure that each output corresponds to exactly one input.
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Ensuring Function Consistency: A function must assign exactly one output to each input. If we consider the whole domain of a trigonometric function, we would have multiple outputs for most values, which violates the definition of a function. By limiting the domain, we ensure that each output has a singular, defined source, making it possible to define a proper inverse.
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Defining the Inverse Function: The primary goal of creating inverse trigonometric functions is to reverse the effect of the original trigonometric function. By restricting the domain (for instance, \(\sin(x)\) is typically defined to be between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\)), we can state that the inverse sine function, \(\arcsin(y)\), will only return one unique angle for each value \(y\) in the range \([-1, 1]\).
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Simplifying the Range of Outputs: By setting domain restrictions, we also control the range of outputs for the inverse functions. This is crucial for practical applications, such as solving triangles or modeling periodic phenomena, where a consistent and manageable range of angles is required.
In summary, domain restrictions for trigonometric functions are essential to ensure uniqueness, maintain function consistency, effectively define the inverse function, and simplify the output range.
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