Problem 47 - University Calculus: Early Transcendentals

Question

Which of the expressions are defined, and which are not? Give reasons for your answers. a. \(\sec ^{-1} 0\) b. \(\sin ^{-1} \sqrt{2}\)

Step-by-Step Solution

Answer

Both expressions are undefined.

Step 1: Understanding the Secant Function

The secant function, \(\sec(x)\), is defined as \(\sec(x) = \frac{1}{\cos(x)}\). Therefore, \(\sec^{-1}(0)\) asks for the angle whose secant is zero. For \(\sec^{-1}(a)\) to be defined, \(a\) must be greater than or equal to 1 or less than or equal to -1, as secant is undefined between -1 and 1. Thus, \(\sec^{-1}(0)\) is undefined.

Step 2: Understanding the Sine Function

The sine inverse function, \(\sin^{-1}(x)\), is defined for values \(-1 \leq x \leq 1\). Since \(\sin^{-1}(\sqrt{2})\) involves \(\sqrt{2}\), which is approximately 1.414 and not within the interval \([-1, 1]\), \(\sin^{-1}(\sqrt{2})\) is undefined.

Key Concepts

Inverse Secant Function

In trigonometry, the inverse secant function, denoted as \(\sec^{-1}(x)\), seeks to find the angle whose secant value is \(x\). Let's recall that secant itself is defined as \(\sec(x) = \frac{1}{\cos(x)}\). To understand the inverse secant function, it is crucial to know that it can only be applied to values outside the interval \((-1, 1)\).

This means that for an inverse secant function to be defined, the input value \(x\) must be greater than or equal to 1 or less than or equal to -1. Why is this the case? Because the cosine function, which is the reciprocal base of secant, only reaches values at these extremes. This is what restricts secant to these domains.

One should remember that trying to find an inverse secant of a value like 0, for example \(\sec^{-1}(0)\), results in an undefined expression. Since 0 lies between -1 and 1, secant is not defined for it, making its inverse also undefined.

In practice, when dealing with \(\sec^{-1}()\), always check the domain of the value first. This ensures the expression is valid.

Inverse Sine Function

The inverse sine function, expressed as \(\sin^{-1}(x)\) or \(\arcsin(x)\), is another fundamental concept in trigonometry. It determines the angle whose sine is \(x\). The sine function itself has a range between -1 and 1.

Consequently, if you are looking for \(\sin^{-1}(x)\), the value \(x\) must fall within the interval \([-1, 1]\). If \(x\) goes beyond this interval, there is simply no real angle whose sine could be such a value, as the sine wave does not reach outside this span.

For instance, the expression \(\sin^{-1}(\sqrt{2})\) is undefined. Since \(\sqrt{2}\), which approximates to 1.414, does not lie within the range \([-1, 1]\), no angle exists that possesses a sine this high.

When using the inverse sine function, always ensure your input respects these domain limitations. This guarantees your results are mathematically valid and real.

Domain Restrictions

Domain restrictions are a crucial part of understanding trigonometric functions and their inverses. They ensure that expressions are mathematically legitimate and correspond to possible real values.

When trigonometric inverses are calculated, consider the function's base values:

• For \(\sec^{-1}(x)\), the domain is \((-\infty, -1] \cup [1, \infty)\).
• For \(\sin^{-1}(x)\), the acceptable domain is \([-1, 1]\).

These domain choices arise from restricting the function to output real numbers, as trigonometric values naturally cycle through positive and negative values.

In practical scenarios such as math problems, ensuring that inputs fall within these valid ranges helps avoid undefined situations or errors. It also aids in making sense of the logical limits of these functions.