Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.
$\csc^{-1}\left(-1\right)$

Verified step by step guidance

1. Understand that \( \csc^{-1}(x) \) is the inverse cosecant function, which gives the angle \( \theta \) such that \( \csc(\theta) = x \).
2. Recall that \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Therefore, \( \csc^{-1}(-1) \) means we are looking for an angle \( \theta \) where \( \sin(\theta) = -1 \).
3. The sine function \( \sin(\theta) \) equals \(-1\) at specific angles. Consider the unit circle: \( \sin(\theta) = -1 \) at \( \theta = \frac{3\pi}{2} \) (or \( 270^\circ \)).
4. Verify that \( \theta = \frac{3\pi}{2} \) is within the range of the inverse cosecant function. The principal range for \( \csc^{-1}(x) \) is \([-\frac{\pi}{2}, \frac{\pi}{2}] \) excluding \( 0 \), but for negative values, we consider angles in the third and fourth quadrants.
5. Conclude that the angle \( \theta = \frac{3\pi}{2} \) satisfies the condition \( \csc(\theta) = -1 \), and thus \( \csc^{-1}(-1) = \frac{3\pi}{2} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, arccos, and arccsc, are the functions that reverse the action of the corresponding trigonometric functions. For example, if y = sin(x), then x = arcsin(y). These functions are defined for specific ranges to ensure they are one-to-one, allowing for unique outputs for each input.

Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). It is important to note that csc(x) is undefined where sin(x) = 0. The cosecant function is particularly relevant when evaluating expressions involving inverse cosecant, such as csc^{-1}(-1).

Range of Inverse Cosecant

The range of the inverse cosecant function, csc^{-1}(x), is limited to the intervals (-∞, -1] and [1, ∞). This means that csc^{-1}(x) can only yield values outside the interval (-1, 1), which is crucial when evaluating expressions like csc^{-1}(-1), as it indicates the specific angle whose cosecant is -1.