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(cot⁻¹ 9π)

Verified step by step guidance

1. Identify the expression: \( \cot(\cot^{-1}(9\pi)) \).
2. Recall the property of inverse functions: \( f(f^{-1}(x)) = x \) for all \( x \) in the domain of \( f^{-1} \).
3. Recognize that \( \cot^{-1}(x) \) is the inverse of the cotangent function, meaning \( \cot(\cot^{-1}(x)) = x \) for all \( x \) in the domain of \( \cot^{-1} \).
4. Determine the domain of \( \cot^{-1}(x) \), which is all real numbers.
5. Since \( 9\pi \) is a real number, apply the inverse property: \( \cot(\cot^{-1}(9\pi)) = 9\pi \).

Key Concepts

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other. For a function f and its inverse f⁻¹, the property f(f⁻¹(x)) = x holds true 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 that applying a function and then its inverse returns the original input, which is crucial for evaluating expressions involving inverse trigonometric functions.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cot⁻¹, sec⁻¹, and csc⁻¹, are used to find angles when given a trigonometric ratio. For example, cot⁻¹(x) gives the angle whose cotangent is x. Understanding these functions is essential for solving problems that involve finding angles from given trigonometric values, as seen in the expression cot(cot⁻¹(9π)).

Cotangent Function

The cotangent function, denoted as cot(x), is the reciprocal of the tangent function, defined as cot(x) = 1/tan(x) or cot(x) = cos(x)/sin(x). It is important to recognize that cot(cot⁻¹(x)) simplifies directly to x, provided x is within the appropriate range of the cotangent function. This property is key to evaluating expressions involving cotangent and its inverse.