Problem 72 - Precalculus: Functions and Graphs

Question

Make the trigonometric substitution $$x=a \tan \theta \quad \text { for }-\pi / 2<\theta<\pi / 2 \text { and } a>0.$$ Simplify the resulting expression. $$\frac{\left(x^{2}+a^{2}\right)^{3 / 2}}{x}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{a^3}{x}\).

Step 1: Make the Substitution

We begin by substituting the given trigonometric substitution into the expression. Here, we have \(x = a \tan \theta\). Substitute this into the expression \(\frac{(x^2 + a^2)^{3/2}}{x}\).

Step 2: Simplify the Expression for \(x^2 + a^2\)

Substituting \(x = a \tan \theta\), we get \(x^2 = (a \tan \theta)^2 = a^2 \tan^2 \theta\). Adding this to \(a^2\), we have: \[ x^2 + a^2 = a^2 \tan^2 \theta + a^2 = a^2(\tan^2 \theta + 1) \] Using the identity \(\tan^2 \theta + 1 = \sec^2 \theta\), this simplifies to: \[ x^2 + a^2 = a^2 \sec^2 \theta \]

Step 3: Apply the Power Rule

Replace \(x^2 + a^2\) with \(a^2 \sec^2 \theta\) in the original expression: \[ \frac{((a^2 \sec^2 \theta)^{3/2})}{x} = \frac{((a^3 (\sec^3 \theta)))}{x} \] This simplifies the numerator to: \[ a^3 \sec^3 \theta \]

Step 4: Substitute for \(x\) Again

We still have \(x = a \tan \theta\) in the denominator. Substitute again to get: \[ \frac{a^3 \sec^3 \theta}{a \tan \theta} = \frac{a^2 \sec^3 \theta}{\tan \theta} \]

Step 5: Simplify with Trigonometric Identities

Applying the identity \(\sec \theta = \frac{1}{\cos \theta}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) in the simplified expression: \[ \frac{a^2 \left(\frac{1}{\cos^3 \theta}\right)}{\left(\frac{\sin \theta}{\cos \theta}\right)} = \frac{a^2}{\cos^2 \theta \sin \theta} \] This simplifies further to: \[ \frac{a^2 \cos \theta}{\sin \theta} \] Which is: \[ a^2 \cot \theta \]

Step 6: Return to Original Variables If Needed

If you need the expression in terms of \(x\) instead of \(\theta\), remember that \(\cot \theta = \frac{1}{\tan \theta} = \frac{a}{x}\). So, \(a^2 \cot \theta\) becomes: \[ a^2 \frac{a}{x} = \frac{a^3}{x} \]

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in solving many mathematical problems, especially those involving trigonometric functions. In this exercise, we use some of these identities to simplify expressions efficiently.

First, we consider the identity:

• Pythagorean Identity: \( \tan^2 \theta + 1 = \sec^2 \theta \).

The Pythagorean identity helps us convert the expression \( x^2 + a^2 = a^2 \tan^2 \theta + a^2 \) into \( a^2 \sec^2 \theta \).

Additionally, during simplification, we apply:

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

These identities allow us to manipulate and simplify complex trigonometric expressions to a form that's easier to work with. Understanding how these identities fit together is key to mastering trigonometric substitution problems.

Simplification

Simplification is all about making an expression more manageable and often involves breaking down complex terms into simpler parts. In the exercise provided, we simplified the complex expression \( \frac{((a^2 \sec^2 \theta)^{3/2})}{a \tan \theta} \).

We begin by focusing on the numerator and applied the exponent rule to get \( a^3 \sec^3 \theta \), reducing the radicals' complexity.
Using the identities mentioned previously, we further simplify:

• Dividing out common terms, such as \( a \tan \theta \) in the denominator.
• Converting all terms into sine and cosine to leverage basic trigonometric rules, which results in the simpler expression \( a^2 \cot \theta \).

Simplification involves appreciating when and how to substitute and manipulate parts of the equation without changing the expression's value, aiming to achieve a more straightforward equivalent form.

Precalculus Problem Solving

Precalculus problem solving involves bridging the gap between algebra, geometry, and calculus, often requiring a strong command over trigonometric concepts and identities.

In this exercise, we tackled the problem using trigonometric substitution: a very powerful technique for integrating or simplifying expressions involving radicals or complicated functions. Trigonometric substitutions replace variables with trigonometric functions, leveraging their known identities to simplify the problem.

The substitution \( x = a \tan \theta \) sets the stage for us to use and interconnect with the trigonometric identities, such as \( \sec^2 \theta \) and \( \tan^2 \theta \). This interconnectedness plays a crucial role in precalculus problem solving, as it allows for converting complicated expressions into simpler forms for easy handling.

By working through such exercises, students improve their ability to see these connections and develop problem-solving skills that are crucial for success in more advanced mathematics, such as calculus. Emphasizing practice, understanding the logic behind each step, and mastering the use of identities are key to becoming proficient in precalculus problem solving.