In Exercises 55–62, use the properties of inverse functions f(f⁻¹ (x)) = x for all x in the domain of f⁻¹ and f⁻¹(f(x)) for all x in the domain of f, as well as the definitions of the inverse cotangent, cosecant, and secant functions, to find the exact value of each expression, if possible. cot⁻¹ (cot 3π/4)

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Recall the property of inverse functions: for any function \( f \) and its inverse \( f^{-1} \), \( f^{-1}(f(x)) = x \) holds true for all \( x \) in the domain of \( f \).

Identify the function and its inverse in the expression: here, \( \cot^{-1} \) is the inverse cotangent function, and \( \cot \) is the cotangent function.

Understand the domain and range of the inverse cotangent function: \( \cot^{-1}(x) \) typically returns values in the interval \( (0, \pi) \).

Since \( \cot^{-1}(\cot(\theta)) = \theta \) only if \( \theta \) is in the principal range of \( \cot^{-1} \), check if \( 3\pi/4 \) lies within \( (0, \pi) \).

Because \( 3\pi/4 \) is within \( (0, \pi) \), you can conclude that \( \cot^{-1}(\cot(3\pi/4)) = 3\pi/4 \).

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Inverse Trigonometric Functions

Inverse trigonometric functions reverse the effect of their corresponding trigonometric functions, returning an angle from a given ratio. For example, cot⁻¹(x) gives the angle whose cotangent is x. Understanding their domains and ranges is crucial to correctly evaluate expressions involving these inverses.

Properties of Inverse Functions

The key property f(f⁻¹(x)) = x holds for all x in the domain of f⁻¹, and f⁻¹(f(x)) = x for all x in the domain of f. This means applying a function and its inverse in succession returns the original input, but only when the input lies within the appropriate domain or range.

Principal Values and Domain Restrictions

Inverse trigonometric functions have restricted ranges (principal values) to ensure they are functions. For cot⁻¹(x), the principal value range is typically (0, π). When evaluating cot⁻¹(cot θ), the result is the principal value angle equivalent to θ within this range, not necessarily θ itself.