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)

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 value, which is crucial for solving problems 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. These functions have specific ranges to ensure they are single-valued, which is essential for determining exact values in trigonometric expressions.

Cotangent Function

The cotangent function, defined as cot(θ) = 1/tan(θ) or cot(θ) = cos(θ)/sin(θ), is the reciprocal of the tangent function. Understanding the cotangent function is vital for evaluating expressions like cot⁻¹(cot(3π/4)), as it helps to determine the angle corresponding to a given cotangent value. The angle 3π/4 is in the second quadrant, where cotangent is negative, influencing the output of the inverse function.