Problem 75 - Precalculus: Functions and Graphs

Question

Make the trigonometric substitution $$x=a \sec \theta \quad \text { for } 0<\theta<\pi / 2 \text { and } a>0.$$ Simplify the resulting expression. $$x^{3} \sqrt{x^{2}-a^{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \(a^4 \sec^3 \theta \tan \theta\).

Step 1: Substitution

Replace the variable \(x\) in the given expression \(x^3 \sqrt{x^2 - a^2}\) with the trigonometric substitution \(x = a \sec \theta\). This gives us \((a \sec \theta)^3 \sqrt{(a \sec \theta)^2 - a^2}\).

Step 2: Simplify the Expression

Calculate \((a \sec \theta)^3 = a^3 \sec^3 \theta\). For \((a \sec \theta)^2 - a^2\), it simplifies to \(a^2 \sec^2 \theta - a^2\). Factor \(a^2\) out to get \(a^2 (\sec^2 \theta - 1)\).

Step 3: Use Trigonometric Identity

Apply the trigonometric identity \(\sec^2 \theta - 1 = \tan^2 \theta\). Therefore, \(a^2 (\sec^2 \theta - 1) = a^2 \tan^2 \theta\).

Step 4: Simplify the Entire Expression

The original expression now becomes \(a^3 \sec^3 \theta \cdot \sqrt{a^2 \tan^2 \theta}\). Simplify the square root: \(\sqrt{a^2 \tan^2 \theta} = a \tan \theta\).

Step 5: Combine Terms

The expression \(a^3 \sec^3 \theta \cdot a \tan \theta\) becomes \(a^4 \sec^3 \theta \tan \theta\).

Step 6: Final Simplified Expression

Thus, the simplified expression after substitution and simplification is \(a^4 \sec^3 \theta \tan \theta\).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical relationships between different trigonometric functions. They often help in transforming complex expressions into simpler or alternative forms. One of the fundamental identities used in trigonometric substitution is the Pythagorean identity:

• It states that \[ \sec^2 \theta = 1 + \tan^2 \theta \]

Understanding and applying these identities allow mathematicians to rewrite expressions in more manageable forms. In the original exercise, we used the identity \(\sec^2 \theta - 1 = \tan^2 \theta\) to simplify the expression post substitution.
This particular identity helps to relate secant and tangent in terms of squares, providing an easier path forward when simplifying radical expressions involving trigonometric functions.
Trigonometric identities are immensely useful in calculus, physics, and engineering, where they provide elegant solutions to otherwise intricate problems.

Simplifying Expressions

Simplifying expressions is a crucial skill in mathematics, reducing complicated equations or formulas into more understandable forms. This process typically involves substituting values, reorganizing terms, or eliminating redundancies.
In the context of this exercise, two significant steps were involved in the simplification process:

• First, substituting \( x = a \sec \theta \) transformed the original expression \( x^3 \sqrt{x^2 - a^2} \) into something manageable.
• Second, leveraging trigonometric identities and basic algebra, particularly factoring and applying square roots, further reduced the expression to its final simplified form.

Simplifying expressions ensures that the essential characteristics of the expression remain, while making computations easier. It is an indispensable skill for anyone studying mathematics or related fields, as it is foundational for solving complex equations efficiently.

Secant and Tangent Functions

The secant and tangent functions are essential components of trigonometry, especially in the scenario of trigonometric substitution. The secant function, \( \sec \theta = \frac{1}{\cos \theta} \), is the reciprocal of cosine. Meanwhile, the tangent function, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), which measures the steepness of an angle, often relates angles on a unit circle.
In trigonometric substitution, the secant and tangent functions are particularly favored due to how they simplify expressions involving square roots. In the exercise, replacing \( x \) with \( a \sec \theta \) and using relevant identities allowed us to use the relationship between secant and tangent.

• This substitution helps convert complexity into simplicity, such as transforming square root expressions into more straightforward tangent squared terms.

Understanding the interplay between these functions is critical for effective precalculus problem-solving, particularly when simplifying complex polynomial or radical expressions.

Precalculus Problem-Solving

Precalculus problem-solving often requires understanding various mathematical concepts, such as trigonometry, algebra, and geometry. The process involves breaking down complex problems into simpler parts, applying mathematical principles, and logically combining results.

• In this exercise, recognizing where a trigonometric substitution was useful turned a challenging expression into a manageable one by introducing a parameter \( \theta \).
• It requires familiarity with manipulative techniques, like factoring and using identities. This lays the groundwork for further exploration in calculus.

Precalculus problem-solving emphasizes strategic thinking and methodical application of known techniques to reach a solution. Mastery of these skills provides a solid foundation for calculus and other advanced mathematical studies, equipping students with the tools needed to tackle various mathematical challenges effectively.