Problem 91 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem Statement

We are given the expression \( \frac{x}{\sqrt{1-x^{2}}} \) with the substitution \( x = \sin \theta \). We need to simplify this expression with the given substitution.

Step 2: Substitute the Given Value

Substitute \( x = \sin \theta \) into the expression. This yields: \[ \frac{\sin \theta}{\sqrt{1-(\sin \theta)^{2}}} \].

Step 3: Simplify the Denominator

Use the trigonometric identity \( 1 - \sin^2 \theta = \cos^2 \theta \) to simplify the denominator. Thus, the expression becomes: \[ \frac{\sin \theta}{\sqrt{\cos^2 \theta}} \].

Step 4: Simplify the Expression

Since \( \sqrt{\cos^2 \theta} = \cos \theta \) when \( 0 \leq \theta < \frac{\pi}{2} \), the expression simplifies to: \[ \frac{\sin \theta}{\cos \theta} \].

Step 5: Use Trigonometric Identity

Recognize that \( \frac{\sin \theta}{\cos \theta} \) is equivalent to \( \tan \theta \). Thus, the simplified expression is \( \tan \theta \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships between the trigonometric functions that are true for all values of the angles involved. They help in simplifying expressions and solving equations in trigonometry. A common identity used in this exercise is the Pythagorean identity:

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

These identities simplify the expressions by transforming one function into another related function using basic algebraic manipulations. Knowing these identities by heart can greatly assist in solving trigonometric problems quickly.
In our problem, we used the identity to replace \(1 - \sin^2 \theta\) with \(\cos^2 \theta\). This is a key step for simplifying the denominator of the given expression.

Simplifying Expressions

Simplifying mathematical expressions involves rewriting them in a simpler or more efficient form without changing their value. This often involves eliminating complex radicals, combining like terms, or using known identities as shortcuts.For this exercise, after substituting \(x = \sin \theta\), we focused on simplifying the denominator \(\sqrt{1 - \sin^2 \theta}\). Using the trigonometric identity \(1 - \sin^2 \theta = \cos^2 \theta\), the expression inside the square root became \(\cos^2 \theta\), allowing us to further simplify to \(\cos \theta\) since the set conditions ensure non-negative values.
Ultimately, the goal is to make the expression easier to work with by expressing it in terms of fewer or simpler functions.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are used to relate the angles of a triangle to the lengths of its sides. They are essential tools in various areas of mathematics and physics.In our problem, we started with \(x = \sin \theta\) and transformed the expression \(\frac{x}{\sqrt{1-x^{2}}}\) accordingly. The sine function describes the ratio of the opposite side to the hypotenuse in a right triangle. When \(x = \sin \theta\), this substitution reflects that meaningful relationship in a given angle.
Through simplification, the expression naturally converted to the trigonometric function \(\tan \theta\), where tangent is the ratio of sine to cosine, given as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent mathematical relationships. In trigonometric substitution problems, these expressions are transformed using trigonometric identities.Initially, we started with the algebraic expression \(\frac{x}{\sqrt{1-x^{2}}}\). By applying the substitution and simplifying techniques, the expression reformed into \(\tan \theta\). This transition highlights how algebraic expressions can be manipulated under different mathematical contexts.
Trigonometric substitution is particularly useful for dealing with expressions involving square roots of quadratic terms, where direct simplification is not straightforward. Recognizing equivalent trigonometric forms allows for efficient and effective simplification or integration of such expressions, often making algebraic manipulations more approachable.