Problem 67 - Precalculus: Functions and Graphs

Question

Make the trigonometric substitution \(x=a \sin \theta\) for \(-\pi / 2<\theta<\pi / 2\) and \(a>0 .\) Use fundamental identities to simplify the resulting expression. $$\frac{x^{2}}{\sqrt{a^{2}-x^{2}}}$$

Step-by-Step Solution

Answer

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

Step 1: Perform the Trigonometric Substitution

Given the substitution \(x = a \sin \theta\), we need to express \(x^2\) and \(a^2 - x^2\) in terms of \(\theta\). Substitute \(x\) in the given expression with \(a \sin \theta\). So, \(x^2 = (a \sin \theta)^2 = a^2 \sin^2 \theta\).

Step 2: Simplify the Denominator

In the denominator, substitute \(x = a \sin \theta\) into \(\sqrt{a^2 - x^2}\). We have: \[\sqrt{a^2 - x^2} = \sqrt{a^2 - a^2 \sin^2 \theta} = \sqrt{a^2 (1 - \sin^2 \theta)} = \sqrt{a^2 \cos^2 \theta} = a \cos \theta\].

Step 3: Substitute and Simplify the Full Expression

Substitute the results from Step 1 and 2 back into the original expression: \[\frac{x^2}{\sqrt{a^2-x^2}} = \frac{a^2 \sin^2 \theta}{a \cos \theta} = a \frac{\sin^2 \theta}{\cos \theta}\].

Step 4: Simplify Further Using Trigonometric Identity

The expression \(a \frac{\sin^2 \theta}{\cos \theta}\) can be further simplified using trigonometric identity: \(\sin^2 \theta = 1 - \cos^2 \theta\). Therefore:\[a \frac{\sin^2 \theta}{\cos \theta} = a \frac{1 - \cos^2 \theta}{\cos \theta} = a \left( \frac{1}{\cos \theta} - \cos \theta \right) = a (\sec \theta - \cos \theta)\].

Key Concepts

Trigonometric Identities

Trigonometric identities are powerful tools in mathematics that relate the angles and sides of triangles. They are crucial for simplifying expressions, particularly when dealing with trigonometric substitutions. In the given exercise, we first introduced the identity for sine: \(x = a \sin \theta\). This defines the relationship between the angle \(\theta\) and the variable \(x\), standard in trigonometric substitution problems.
The identity \(\sin^2 \theta + \cos^2 \theta = 1\) is pivotal in this context because it allows us to transform expressions into simpler forms through substitution. For example, substituting \(x = a \sin \theta\) enables us to express \(x^2\) as \(a^2 \sin^2 \theta\).
Moreover, the problem highlights the use of the Pythagorean identity to simplify the square root expression in the denominator: \(\sqrt{a^2(1-\sin^2 \theta)} = a \cos \theta\).

• Remember: \(\sin^2 \theta = 1 - \cos^2 \theta\), and vice versa.
• This identity is essential for rewriting expressions without involving square roots, which simplifies further calculations.

Simplifying Expressions

Simplifying expressions is a key skill in precalculus and calculus to make complex equations more manageable and solvable. In this exercise, after performing the trigonometric substitution \(x = a \sin \theta\), the subsequent steps focus on simplifying both the numerator and the denominator of the expression.
First, we write \(x^2\) as \(a^2 \sin^2 \theta\). We then substitute this into the original expression to form the numerator. The denominator, initially intricate, becomes easier to handle by recognizing that \(\sqrt{a^2 - x^2} = a \cos \theta\). These transformations are possible because of the trigonometric identities used.
After substituting, the expression becomes: \(\frac{a^2 \sin^2 \theta}{a \cos \theta}.\) This part seems straightforward, yet it requires precision: canceling common factors and recognizing the new form. After canceling \(a\), our expression simplifies to \(a \frac{\sin^2 \theta}{\cos \theta}.\)To simplify further, we apply identities:

• Break up the expression using \(\sin^2 \theta = 1 - \cos^2 \theta\).
• The expression transforms to \(a(\sec \theta - \cos \theta)\).

Precalculus Mathematics

Precalculus mathematics lays the groundwork for understanding limits, derivatives, and integrals in calculus. It encompasses a broad range of topics, including algebra, functions, and trigonometry. The exercise provided is typical of precalculus tasks that lead students to discover algebraic manipulations alongside trigonometric concepts.
One essential idea here is mastering substitutions. Substitution methods strengthen a student's ability to change variables and find new ways to approach solving an equation. By substituting \(x = a \sin \theta\), you transform an algebraic expression into a trigonometric one, opening doors to simpler calculations and deeper insights.
The substitution introduces a new variable, making the expression easier to handle. This process showcases the transition from abstract algebra to more tangible trigonometric functions and highlights the seamless blend of previously learned mathematical fields in precalculus.
In practice, mastering these skills prepares students for more advanced studies in calculus. Here, they will engage with more sophisticated forms of these expression manipulations, further expanding their mathematical toolkit.