Problem 93 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\( \tan \theta \)

Step 1: Understand the Substitution

We start with the expression \( \sqrt{x^2 - 1} \) and know that \( x = \sec \theta \). Therefore, \( x^2 = (\sec \theta)^2 \). Since \( \sec \theta = \frac{1}{\cos \theta} \), then \( \sec^2 \theta = 1 + \tan^2 \theta \) by the identity \( \sec^2 \theta = 1 + \tan^2 \theta \).

Step 2: Substitute and Simplify

Substitute \( x^2 \) in the expression: \( \sqrt{x^2 - 1} = \sqrt{\sec^2 \theta - 1} \). Using the identity \( \sec^2 \theta = 1 + \tan^2 \theta \), we have \( \sqrt{\tan^2 \theta} \). Since \( \tan \theta \geq 0 \) for \( 0 \leq \theta < \pi/2 \), this simplifies to \( \tan \theta \).

Step 3: Final Expression

The simplified form of the given expression is \( \tan \theta \) as derived from the trigonometric identity and given substitution.

Key Concepts

Trigonometric Identities

Trigonometric identities are key tools in simplifying expressions and solving equations in trigonometry. They are equations that relate the trigonometric functions, such as sine, cosine, and tangent, to one another. These identities are derived based on the right triangle definitions or the unit circle approach. They allow us to transform complex trigonometric expressions into simpler forms. For example, one crucial identity often used is

• the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\)

Another useful set of identities involves secant and tangent functions, such as:

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

This particular identity is a rearrangement of the Pythagorean identity for practical use when working with secant and tangent functions. Using these identities, we can simplify expressions by replacing one function with another, which can help in rationalizing or integrating expressions.

Secant Function

The secant function, denoted as \(\sec \theta\), is one of the six primary trigonometric functions. It is defined as the reciprocal of the cosine function:

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

The secant function is particularly useful in cases where the angle \(\theta\) corresponds to positions on the unit circle where the cosine value is non-zero. As it is derived from cosine, \(\sec \theta\) assumes values in the range of negative to positive infinity, excluding the values where \(\cos \theta = 0\).
In the original exercise, the substitution \(x = \sec \theta\) simplifies solving methods by transforming the square root expression \(\sqrt{x^2 - 1}\) into a more manageable trigonometric form. This use of the secant function aligns with its property in trigonometric identities, especially with one involving tangent function \(\sec^2 \theta = 1 + \tan^2 \theta\).

Tangent Function

The tangent function, expressed as \( \tan \theta \), is another primary trigonometric function, known for the ratio it offers between sine and cosine:

• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

This definition connects \(\tan \theta\) with the other trigonometric functions and provides a path to direct application in problem solving.
In the range \(0 \leq \theta < \pi/2\), the tangent function is positive and increases in value as \(\theta\) approaches \(\pi/2\). This makes it a stable choice for many substitutions involving expressions like \(\sqrt{x^2 - 1}\), often simplifying into \(\tan \theta\).
The substitution \(x = \sec \theta\) allows us to utilize the identity \(\sec^2 \theta = 1 + \tan^2 \theta\), transforming \(\sqrt{x^2 - 1}\) into \(\sqrt{\tan^2 \theta}\). Because \(\tan \theta\) is non-negative in this specific domain, it directly simplifies to \(\tan \theta\), showcasing the practical side of trigonometric identities when efficiently substituting into algebraic expressions.