Find the exact value of each real number y if it exists. Do not use a calculator.
y = csc⁻¹ (―2)

Key Concepts

Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function. It is defined as csc(x) = 1/sin(x). The cosecant function is only defined for values of x where sin(x) is not zero, which occurs at integer multiples of π. Understanding this function is crucial for solving problems involving its inverse, csc⁻¹.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as csc⁻¹(x), are used to find angles whose cosecant is a given value. The range of csc⁻¹(x) is limited to specific intervals to ensure it is a function, typically [−π/2, 0) ∪ (0, π/2]. This concept is essential for determining the angle corresponding to a given cosecant value.

Domain and Range of Inverse Functions

The domain of the inverse cosecant function, csc⁻¹(x), is x ≤ -1 or x ≥ 1, meaning it only accepts values outside the interval (-1, 1). The range, as mentioned, is restricted to angles in the first and fourth quadrants. Recognizing these constraints is vital for determining whether a solution exists for a given input, such as -2 in this case.