Problem 73 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 73-76: Make the trigonometric substitution $$ x=a \sec \theta \text { for } 0<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ \sqrt{x^{2}-a^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( a \tan \theta \).

Step 1: Understand the Substitution

We are asked to use the trigonometric substitution \( x = a \sec \theta \). This implies that \( \theta \) is such that \( \sec \theta = \frac{x}{a} \) and \( a \cos \theta = \frac{a}{x} \). Our goal is to substitute \( x \) in the expression \( \sqrt{x^2 - a^2} \) with \( a \sec \theta \).

Step 2: Substitute in the Expression

Substitute \( x = a \sec \theta \) into \( \sqrt{x^2 - a^2} \). We get:\[ \sqrt{(a \sec \theta)^2 - a^2} \] Simplify inside the square root:\[ \sqrt{a^2 \sec^2 \theta - a^2} \]

Step 3: Factor the Expression

Factor the expression inside the square root:\[ \sqrt{a^2(\sec^2 \theta - 1)} \] This uses the Pythagorean identity for secant, \( \sec^2 \theta - 1 = \tan^2 \theta \).

Step 4: Simplify using Trigonometric Identity

Apply the trigonometric identity based on the previous step:\[ \sqrt{a^2 \tan^2 \theta} \]This simplifies further to:\[ a |\tan \theta| \]

Step 5: Evaluate Absolute Value

Since \( 0 < \theta < \pi/2 \), \( \tan \theta \) is positive, so the absolute value can be removed, giving:\[ a \tan \theta \] as the final simplified expression.

Key Concepts

Pythagorean Identities

Pythagorean Identities are essential tools in trigonometry used to relate the functions sine, cosine, and tangent to one another. One of the most popular and useful identities is

• \(\sin^2 \theta + \cos^2 \theta = 1\)

From this, we can derive other identities, such as for the secant and tangent functions:

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

This identity is particularly helpful when working with expressions involving the secant function. By substituting \(\sec^2 \theta - 1\) with \(\tan^2 \theta\), we can simplify complex trigonometric expressions more easily.
In this particular exercise, this identity was pivotal to simplifying the square root expression \(\sqrt{a^2(\sec^2 \theta - 1)}\) into \(\sqrt{a^2 \tan^2 \theta}\). Expanding upon these identities can make tackling trigonometric problems far more straightforward.

Trigonometric Simplification

Trigonometric Simplification involves reducing complex trigonometric expressions to simpler forms. This process often leverages known identities, like the Pythagorean identities, to replace parts of a trigonometric expression with more manageable equivalents.
For our problem involving the trigonometric substitution \(x = a \sec \theta\), simplification started with replacing \(x\) using this substitution, resulting in \(\sqrt{(a \sec \theta)^2 - a^2}\). Then, recognizing the identity \(\sec^2 \theta - 1 = \tan^2 \theta\), we were able to simplify \(\sqrt{a^2 (\sec^2 \theta - 1)}\) further to \(a \tan \theta\).
Effective simplification makes use of absolute values, especially in determined domains for angle \(\theta\). In this exercise, because \(0 < \theta < \pi/2\) means \(\tan \theta\) is positive, the absolute value could be dropped immediately. This attention to detail helps ensure the final simplification accurately reflects the problem's constraints.

Secant Function

The Secant Function, denoted as \(\sec \theta\), is one of the six main trigonometric functions and is the reciprocal of the cosine function:

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

This function relationships help with transforming expressions and tackling integrals or derivatives involving trigonometric functions.
In the context of trigonometric substitution, using \(x = a \sec \theta\) provides a way to tackle expressions like \(\sqrt{x^2 - a^2}\). By substituting, we express \(x\) in terms of angle \(\theta\) and scrutinize how this changes the expression.
Recognizing that \(\theta\) belongs to the interval \(0 < \theta < \pi/2\), we know \(\cos \theta\) and consequently \(\sec \theta\) are both positive, simplifying further manipulation of the expression. This approach helps transform complex algebraic expressions into simpler trigonometric forms, paving the way for easier evaluation or integration.