Problem 46 - University Calculus: Early Transcendentals

Question

Which of the expressions are defined, and which are not? Give reasons for your answers. a. \(\csc ^{-1}(1 / 2)\) b. \(\csc ^{-1} 2\)

Step-by-Step Solution

Answer

a. Undefined; b. Defined.

Step 1: Understanding the Function Domain

The function \(\csc^{-1}(x)\) is the inverse of the cosecant function. The cosecant function is defined as \(\csc(\theta) = \frac{1}{\sin(\theta)}\), and it is undefined where \(\sin(\theta) = 0\). However, for the inverse functions, we look at the range of cosecant values from -∞ to -1 and 1 to ∞, as \(\csc^{-1}(x)\) is only defined for these ranges.

Step 2: Analyzing Expression a

The expression \(\csc^{-1}(1/2)\) requires that \(1/2\) lie within the range of values where \(\csc^{-1}(x)\) is defined, i.e., \(-\infty < x \leq -1\) or \(1 \leq x < \infty\). Since \(1/2\) does not meet this criteria, \(\csc^{-1}(1/2)\) is not defined.

Step 3: Analyzing Expression b

Similarly, checking \(\csc^{-1}(2)\), the number 2 lies within the range where \(\csc^{-1}(x)\) is defined, 1 to ∞. Hence, \(\csc^{-1}(2)\) is defined.

Key Concepts

Cosecant Function

The cosecant function is an important player in the family of trigonometric functions. It is the reciprocal of the sine function, essentially meaning that for any angle \( \theta \), the cosecant of \( \theta \) is \( \csc(\theta) = \frac{1}{\sin(\theta)} \). This sets the stage for understanding how it behaves differently from the more frequently discussed sine wave.
The cosecant function is noted for its undefined points wherever the sine function equals zero, since division by zero is undefined. As the sine of an angle approaches zero, the cosecant function soars towards positive or negative infinity, making it especially intriguing yet challenging for students to work with.
The inverse, \( \csc^{-1}(x) \), lets us determine the angle \( \theta \) when we already know the cosecant value. But, unlike sine and cosine which are defined anywhere on the number line from -1 to 1, the domain for \( \csc(\theta) \) is different, leading us to our next topic: function domains.

Function Domains

Understanding function domains is essential to tackling any trigonometric problem effectively. For \( \csc^{-1}(x) \), domain analysis is crucial. Unlike some functions, which can accept any real number, the cosecant function operates in two separate ranges: from \(-\infty\) to \(-1\) and from \(1\) to \(\infty\).
This means that \( \csc^{-1}(x) \) is only defined for values outside of the open interval from \(-1\) to \(1\). Hence, if an input, such as \( \frac{1}{2} \), lies between \(-1\) and \(1\), \( \csc^{-1}(x) \) cannot produce a valid output. For instance, \( \csc^{-1}(\frac{1}{2}) \) is not defined since \( \frac{1}{2} \) is in that restricted region.

\( \csc^{-1}(2) \) is a valid expression because 2 falls within the permissible domain of \(1 \leq x < \infty\).

This domain restriction is essential knowledge for anyone delving deep into inverse trigonometric functions.

Mathematical Analysis

In mathematical analysis, we break down problems to uncover their underlying structures. With inverse trigonometric functions, the analysis often involves checking where a function is defined and, consequently, where it isn’t.
When analyzing \( \csc^{-1}(1/2) \), we apply what we know about the function's domain. Since \(1/2\) falls within the restricted domain where \( \csc^{-1}(x) \) is not defined, the analysis logically concludes that the expression is undefined.
However, scrutinizing \( \csc^{-1}(2) \) reveals it's a defined expression. This involves seeing that 2 lies in the domain, permitting \( \csc^{-1}(2) \) to provide a meaningful angle result.

Consistent analysis requires verifying function rules and domains.

By carefully scrutinizing inputs and outputs, we grow our understanding of how inverse trigonometric functions operate. This kind of analysis builds a foundational skill set beneficial for tackling complex trigonometry problems in the future.