Problem 69 - 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. $$\sqrt{a^{2}+x^{2}}$$

Step-by-Step Solution

Answer

\(\sqrt{a^{2}+x^{2}} = a \sec \theta\).

Step 1: Identify the substitution

In the given problem, we need to perform a trigonometric substitution. We are told to use the substitution \(x = a \tan \theta\), where \(a > 0\) and \(-\pi/2 < \theta < \pi/2\). This substitution will allow us to simplify the given expression \(\sqrt{a^{2}+x^{2}}\).

Step 2: Substitute \(x = a \tan \theta\) into \(\sqrt{a^{2} + x^{2}}\)

Substitute \(x = a \tan \theta\) into the expression. The expression becomes \(\sqrt{a^{2} + (a \tan \theta)^{2}}\). Simplifying inside the square root gives us \(\sqrt{a^{2} + a^{2} \tan^{2} \theta}\).

Step 3: Factor out \(a^2\) from the square root

Notice the common factor of \(a^{2}\) in the terms inside the square root: \(a^{2} + a^{2} \tan^{2} \theta = a^{2}(1 + \tan^{2} \theta)\). Now, the expression becomes \(\sqrt{a^{2}(1 + \tan^{2} \theta)}\).

Step 4: Use the Pythagorean identity \(1 + \tan^{2} \theta = \sec^{2} \theta\)

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

Step 5: Simplify the expression

Since \(\sqrt{a^{2} \sec^{2} \theta} = \sqrt{a^{2}} \cdot \sqrt{\sec^{2} \theta}\), simplify it to \(a \cdot \sec \theta\). Thus, the simplified form of the initial expression is \(a \sec \theta\).

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental concept in trigonometry. It states that for any angle \( \theta \): \[ 1 + \tan^2 \theta = \sec^2 \theta \].
This identity is derived from the Pythagorean theorem and applies to right-angled triangles, where the hypotenuse is \( \sec \theta \) and the opposite side is \( \tan \theta \).
Knowing this identity allows us to connect and convert different trigonometric functions into one another, leveraging the relationship between tangent and secant.

In our exercise, we replaced \(1 + \tan^2 \theta\) in the expression \(\sqrt{a^{2}(1 + \tan^{2} \theta)}\) with \(\sec^2 \theta\).
This simplification is crucial because it makes the expression easier to work with and reduces it to terms of \( \sec \theta \), simplifying the square root to \(a \cdot \sec \theta\).
The Pythagorean identity is often used to simplify expressions, solve equations, and integrate functions in calculus.

simplifying expressions

Simplifying mathematical expressions is a key skill in mathematics. The goal is to rewrite expressions in their simplest form, often to make them easier to interpret or manipulate.
When simplifying, look for opportunities to factor, distribute, combine like terms, and recognize patterns or identities.

• In the exercise, we first substituted the expression \(x = a \tan \theta\) into the original equation, resulting in \( \sqrt{a^{2} + a^{2} \tan^{2} \theta} \).
• Next, we factored out the common factor \(a^{2}\), giving us \( \sqrt{a^{2}(1 + \tan^{2} \theta)} \).
• Utilizing the identity \(1 + \tan^{2} \theta = \sec^{2} \theta\) further allowed us to convert the expression under the square root to a simpler form.

Each step in simplifying an expression can require different strategies, but with practice, identifying these opportunities becomes second nature. The act of simplification not only aids in solving problems but also deepens understanding of the relationships between mathematical principles.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are universally true for any value of the variables involved. They play a critical role in simplifying expressions, solving trigonometric equations, and proving mathematical statements.
Common identities include the Pythagorean identities, angle sum and difference formulas, and the double angle formulas.

• The Pythagorean identity used in our exercise, \(1 + \tan^2 \theta = \sec^2 \theta\), is just one example. Such identities simplify the analysis of trigonometric expressions.
• Trigonometric identities help convert complex expressions into simpler or more familiar forms, helping us solve equations more effectively.
• Understanding these identities aids in recognizing when and how to apply them, making complex calculus problems more manageable.

By memorizing and practicing these identities, one can greatly enhance their ability to approach and solve a variety of mathematical problems involving trigonometric functions.