Problem 93 - Algebra and Trigonometry

Question

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7\()\) . Assume that \(0 \leq \theta<\pi / 2 .\) $$ \sqrt{x^{2}-1}, \quad x=\sec \theta $$

Step-by-Step Solution

Answer

\( \tan \theta \)

Step 1: Identify the substitution

We need to make the substitution given in the problem. Here, we are told that \( x = \sec \theta \). Let's substitute \( x = \sec \theta \) into the expression \( \sqrt{x^2 - 1} \).

Step 2: Substitute and simplify

Substitute \( x = \sec \theta \) into the expression: \[ \sqrt{(\sec \theta)^2 - 1}. \] According to the identity \( \sec^2 \theta = 1 + \tan^2 \theta \), we can write:\[ \sqrt{\sec^2 \theta - 1} = \sqrt{(1 + \tan^2 \theta) - 1} = \sqrt{\tan^2 \theta}. \]

Step 3: Simplify further

The square root of \( \tan^2 \theta \) simplifies to \( \tan \theta \), using the principal square root in the range \( 0 \leq \theta < \pi / 2 \) since \( \tan \theta \geq 0 \) in this interval.Thus, we have:\[ \sqrt{\tan^2 \theta} = \tan \theta. \]

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is fundamental when substituting algebraic expressions. They serve as powerful tools for transforming and simplifying trigonometric expressions. In this exercise, the identity used is

• \( \sec^2 \theta = 1 + \tan^2 \theta \)

This identity relates the secant and tangent functions, two of the reciprocal trigonometric functions. When substituting \( x = \sec \theta \) into the expression \( \sqrt{x^2 - 1} \), it becomes crucial. The trigonometric substitution allows us to transform our expression into a more manageable form \( \sqrt{\tan^2 \theta} \). It relies heavily on recognizing and applying these identities accurately. Mastering these identities not only aids in simplification but also builds a strong foundation for solving more complex trigonometric problems.

Simplification

Simplification is the process of making an expression easier to understand or work with. Our goal is to convert complicated expressions into their simplest form. This often involves using known identities or algebraic manipulations. Here’s a breakdown of the simplification process in this task:

• Substitute \(x = \sec \theta \) into the expression \(\sqrt{x^2 - 1}\).
• The resulting expression is \(\sqrt{(\sec \theta)^2 - 1}\).
• Using the identity \(\sec^2 \theta = 1 + \tan^2 \theta\), we simplify this to \(\sqrt{\tan^2 \theta}\).

Our expression becomes much simpler - from a complex square root to a simple trigonometric function. This demonstrates the elegance of trigonometric substitution, turning potentially cumbersome expressions into their core trigonometric counterparts. Each simplification step is crucial, leading us closer to a result that's both easy to interpret and use.

Principal Square Root

When dealing with square roots in mathematical expressions, especially in the context of trigonometric functions, understanding the principal square root is key. The principal square root refers to the non-negative square root of a number. It's particularly important when simplifying expressions where the variable is defined over a specific interval, as it ensures the expression's validity.In this context, the principal square root is used in the expression \(\sqrt{\tan^2 \theta}\). Because the interval given is \(0 \leq \theta < \pi/2\), the tangent is non-negative in this range. This allows us to assert confidently that:

• \(\sqrt{\tan^2 \theta} = \tan \theta\)

This holds true because the square of \(\tan \theta\) is positive or zero within the defined interval, aligning with the principal square root's definition. Understanding this assures students that not only are they simplifying correctly, they're also staying true to mathematical definitions and ensuring the expression remains valid across its possible values.