Problem 92 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( \sec \theta \).

Step 1: Identify the Substitution

We start by identifying the substitution given in the problem. We have that the variable \( x \) is expressed in terms of \( \theta \) with the relation \( x = \tan \theta \).

Step 2: Substitute into the Expression

Replace \( x \) with \( \tan \theta \) in the expression \( \sqrt{1 + x^2} \). Thus, we have \( \sqrt{1 + (\tan \theta)^2} \).

Step 3: Use Trigonometric Identity

Recall the trigonometric identity: \( 1 + \tan^2 \theta = \sec^2 \theta \). Substitute this identity into the expression, meaning \( \sqrt{1 + \tan^2 \theta} = \sqrt{\sec^2 \theta} \).

Step 4: Simplify the Expression

Simplify \( \sqrt{\sec^2 \theta} \). Since \( \sec \theta = \frac{1}{\cos \theta} \) and the square root of a square results in the absolute value, we get \( |\sec \theta| \).

Step 5: Consider the Domain

Given that \( 0 \leq \theta < \frac{\pi}{2} \), \( \cos \theta \) is positive, so \( \sec \theta \) is also positive. This means \( |\sec \theta| = \sec \theta \). Thus, \( \sqrt{1 + x^2} = \sec \theta \).

Key Concepts

Trigonometric Identities

Trigonometric identities are like secret tools that help us solve problems involving triangles and angles. These identities show the relationships between different trigonometric functions. One of the most useful identities is the Pythagorean Identity. It states:

• \(\sin^2 \theta + \cos^2 \theta = 1\)
• Derived variations like \(1 + \tan^2 \theta = \sec^2 \theta\) are just as important.

When we know this identity, we can translate and simplify trigonometric expressions. In the given exercise, we used the identity \(1 + \tan^2 \theta = \sec^2 \theta\), simplifying \( \sqrt{1 + \tan^2 \theta} \) to \( \sqrt{\sec^2 \theta} \). This step is crucial as it allows us to connect different parts of a trigonometric expression efficiently.

Trigonometric Functions

Trigonometric functions describe relationships in right-angled triangles. The basic functions are sine, cosine, and tangent. They are defined as:

• Sine \(\sin \theta\): opposite side over hypotenuse
• Cosine \(\cos \theta\): adjacent side over hypotenuse
• Tangent \(\tan \theta\): opposite side over adjacent side

There are also reciprocal functions like cosecant (\(\csc \theta\)), secant (\(\sec \theta\)), and cotangent (\(\cot \theta\)). In this problem, we dealt with tangent (\(\tan \theta\)) and secant (\(\sec \theta\)). The relation \(\sec \theta = \frac{1}{\cos \theta}\) becomes essential when simplifying \(\sqrt{\sec^2 \theta}\). Understanding these functions and their relations is vital for solving trigonometric substitution problems.

Simplifying Expressions

Simplifying expressions makes them easier to work with and understand. This involves using mathematical identities or properties to reduce complex expressions into simpler ones. In the exercise, the expression \(\sqrt{1 + x^2}\) was simplified using the substitution \(x = \tan \theta\). Once substituted, the identity \(1 + \tan^2 \theta = \sec^2 \theta\) converted the expression to \(\sqrt{\sec^2 \theta}\). The key to further simplifying was recognizing that the square root of a square yields an absolute value, \(|\sec \theta|\). Since \(0 \leq \theta < \frac{\pi}{2}\), \(\sec \theta\) is positive, allowing us to say \(\sqrt{\sec^2 \theta} = \sec \theta\). This transformation simplifies the expression to \(\sec \theta\) without losing any meaning. Mastering simplification techniques enhances problem-solving skills, particularly in calculus and trigonometry.