Problem 95 - Precalculus: Functions and Graphs

Question

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta .\) $$\sqrt{1+\tan ^{2} \theta} ; \quad 3 \pi / 2<\theta<2 \pi$$

Step-by-Step Solution

Answer

The expression is rewritten as \(\sec \theta\) for the given interval.

Step 1: Understanding the Expression

The expression provided is \( \sqrt{1 + \tan^2 \theta} \). According to a trigonometric identity, \( 1 + \tan^2 \theta = \sec^2 \theta \). This allows us to rewrite the expression in terms of \( \sec \theta \).

Step 2: Substitute the Trigonometric Identity

Using the identity \( 1 + \tan^2 \theta = \sec^2 \theta \), substitute into the expression:\[\sqrt{1 + \tan^2 \theta} = \sqrt{\sec^2 \theta}\]

Step 3: Simplify the Expression

Simplify the square root of a square. Generally, \( \sqrt{a^2} = |a| \). Thus, \( \sqrt{\sec^2 \theta} = |\sec \theta| \).

Step 4: Determine the Sign of \(\sec \theta\) in the Given Interval

We have \( 3\pi/2 < \theta < 2\pi \). In this interval, \( \sec \theta = 1/\cos \theta \) and since \(\cos \theta\) is positive, \(\sec \theta\) is also positive. Therefore, the expression simplifies to \( \sec \theta \).

Step 5: Rewrite the Expression without Absolute Values

Since \(\sec \theta\) is positive within the given interval, we can express the nonradical form as simply \( \sec \theta \) without needing absolute value signs.

Key Concepts

Understanding the Secant Function

When diving into the realm of trigonometric identities, the secant function often emerges as a pivotal concept. The secant function, denoted as \( \sec \theta \), is closely related to the cosine function:

• \( \sec \theta = \frac{1}{\cos \theta} \)

This relationship signifies that wherever the cosine function is non-zero, the secant function will be defined and calculable.
In the context of trigonometric identities, the expression \( \sec^2 \theta \) is especially significant as it can simplify complex trigonometric expressions. For instance, through the identity \( 1 + \tan^2 \theta = \sec^2 \theta \), we can clarify complex expressions like \( \sqrt{1 + \tan^2 \theta} \).
This relationship is invaluable in streamlining problems by substituting \( \sec^2 \theta \) for \( 1 + \tan^2 \theta \), allowing for easier simplification and computation.

Utilizing Interval Notation

Interval notation is a mathematical tool used to describe a range or set of numbers along a continuum. It is especially useful for expressing domains of functions or solutions in a concise way.

• Consider the interval \( (3\pi/2, 2\pi) \). Here, \( \theta \) values lie strictly between these two points.
• Parentheses \( () \) indicate that the endpoints are not included, distinguishing it from brackets \( [] \), which include the endpoints.

Within the specified domain of \( 3\pi/2 < \theta < 2\pi \), the behaviour of trigonometric functions can vary greatly from other intervals. Particularly, during this interval, the cosine function is positive, affecting the secant function such that \( \sec \theta \) remains positive. This insight is crucial when simplifying trigonometric expressions without concern for absolute values because we know the function's behaviour within this interval.

The Art of Simplifying Expressions

Simplifying expressions is a cornerstone of solving mathematical problems, and it involves making an expression as concise as possible while maintaining its original value. Here's how it works in the context of the given problem:

• The original radical expression \( \sqrt{1 + \tan^2 \theta} \) utilizes the trigonometric identity \( 1 + \tan^2 \theta = \sec^2 \theta \).
• Applying this identity, the expression simplifies to \( \sqrt{\sec^2 \theta} \), which mathematically equates to \( |\sec \theta| \).

The subsequent step requires assessing the sign of \( \sec \theta \) over the given interval. Since we determined \( \sec \theta \) to be positive on \( (3\pi/2, 2\pi) \), the absolute value notation can be discarded, resulting in the final simplified form, \( \sec \theta \).
This journey through trigonometric identities, interval understanding, and expression simplification illustrates the interconnectedness of mathematical concepts and the elegance of streamlined solutions.