Problem 69 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 69-72: Make the trigonometric substitution $$ x=a \tan \theta \text { for }-\pi / 2<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ \sqrt{a^{2}+x^{2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(a \sec \theta\).

Step 1: Identify the Substitution

In this problem, we will use the trigonometric substitution \(x = a \tan \theta\). This substitution is suggested to help simplify the given expression \( \sqrt{a^2 + x^2} \). We know from trigonometry that \(\sec^2 \theta = 1 + \tan^2 \theta\).

Step 2: Substitute into the Expression

Substitute \(x = a \tan \theta\) into the expression \(\sqrt{a^2 + x^2}\). This gives us: \(\sqrt{a^2 + (a \tan \theta)^2}\).

Step 3: Simplify Inside the Square Root

Simplify the expression inside the square root: \(\sqrt{a^2 + a^2 \tan^2 \theta}\). Factor out \(a^2\) from the terms: \(\sqrt{a^2(1 + \tan^2 \theta)}\).

Step 4: Use Trigonometric Identity

Apply the trigonometric identity \(1 + \tan^2 \theta = \sec^2 \theta\). This simplifies the expression to \(\sqrt{a^2 \sec^2 \theta}\).

Step 5: Simplify Square Root Expression

The square root of \(a^2 \sec^2 \theta\) is equal to \(a \sec \theta\). Hence, \(\sqrt{a^2 \sec^2 \theta} = a \sec \theta\). Since \(a > 0\) and \(\sec \theta > 0\) within the given range, the expression simplifies to \(a \sec \theta\).

Key Concepts

Trigonometric Identities

Trigonometric identities are crucial tools in simplifying complex mathematical expressions. These identities relate different trigonometric functions to one another. In this exercise, we took advantage of the identity that links tangent and secant:

• The identity used here is: \[1 + \tan^2 \theta = \sec^2 \theta\]

This identity is particularly helpful when dealing with substitution techniques like our current problem's trigonometric substitution.
The function \(\tan \theta\) represents a ratio that originates from right triangle properties, while \(\sec \theta\) is the reciprocal of \(\cos \theta\). These identities serve to connect these angles in a way that makes algebraic simplifications more straightforward. Understanding these identities empowers you to manipulate expressions involving trigonometric functions efficiently.

Simplification of Expressions

Simplifying expressions is a key objective in mathematics. It involves reducing complex mathematical phrases into simpler or more manageable forms. In our exercise, we are tasked with simplifying the expression \(\sqrt{a^2 + x^2}\) using a trigonometric substitution.
First, we replace \(x\) with \(a \tan \theta\), which translates the expression into \( \sqrt{a^2 + (a \tan \theta)^2}\). By expanding and factoring out \(a^2\) inside the square root, the expression becomes:

• \(\sqrt{a^2(1 + \tan^2 \theta)}\)

Recognizing the trigonometric identity within allows further simplification to \(\sqrt{a^2 \sec^2 \theta}\).
The simplification here efficiently uses algebra inside the radical to achieve a final, more straightforward form. Each step reduces the complexity of the expression without altering its value, thus achieving our goal of simplification.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations and expressions to make them easier to solve or simplify. This involves using standard algebraic operations and rules, such as factoring or applying identities. In this trigonometric substitution exercise, algebraic manipulation plays a central role.
Initially, we replace \(x = a \tan \theta\) in the expression \(\sqrt{a^2 + x^2}\). This substitution transforms it into \(\sqrt{a^2 + a^2 \tan^2 \theta}\). By recognizing similar terms, we factor out \(a^2\), resulting in:

• \(\sqrt{a^2(1 + \tan^2 \theta)}\)

Next, using the identity, we substitute \(1 + \tan^2 \theta\) with \(\sec^2 \theta\), leading us to simplify the entire expression to \(\sqrt{a^2 \sec^2 \theta}\).
The final algebraic step involves taking the square root of a product, which simplifies to \(a \sec \theta\). Each algebraic manipulation step is strategic, ensuring the problem is progressively simplified until reaching a workable, simple expression. Mastery of algebraic manipulation ensures tackling more complex problems with confidence and clarity.