Problem 76 - 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. $$\frac{\sqrt{x^{2}-a^{2}}}{x^{2}}$$

Step-by-Step Solution

Answer

\(\frac{\sin \theta}{a}\)

Step 1: Understand Trigonometric Substitution

We are given the substitution \(x = a \sec \theta\). This means we should replace \(x\) with \(a \sec \theta\) in the expression \(\frac{\sqrt{x^{2}-a^{2}}}{x^{2}}\).

Step 2: Substitute x with a sec θ

Replacing \(x\) with \(a \sec \theta\), the expression becomes: \[ \frac{\sqrt{(a \sec \theta)^{2} - a^{2}}}{(a \sec \theta)^{2}} \]

Step 3: Simplify Inside the Square Root

Calculate \((a \sec \theta)^{2} - a^{2}\): \((a \sec \theta)^{2} = a^2 \sec^2 \theta\) which is equal to \(a^2 (\sec^2 \theta - 1)\). Using the identity \(\sec^2 \theta = 1 + \tan^2 \theta\), this becomes \(a^2 \tan^2 \theta\).

Step 4: Simplify the Expression Under the Square Root

Replace \(\sec^2 \theta - 1\) with \(\tan^2 \theta\) to get:\[ \sqrt{a^2 \tan^2 \theta} = a \tan \theta \]

Step 5: Substitute and Simplify the Expression

Replace the expression under the radical in our original fraction:\[ \frac{a \tan \theta}{a^2 \sec^2 \theta} \]This can be simplified to:\[ \frac{a \sin \theta / \cos \theta}{a^2 (1/\cos^2 \theta)} \] or further simplified to :\[ \frac{\sin \theta}{a \cos \theta} \]

Step 6: Final Simplification

Now simplify by multiplying the top and bottom by \(\cos \theta\) to remove the \(\cos\) from the denominator:\[ \frac{\sin \theta}{a} \]

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical expressions that involve trigonometric functions. They are true for any angle and provide a way to simplify and solve trigonometric equations. In this exercise, the primary trigonometric identity we use is:

• The Pythagorean identity: \( \sec^2 \theta = 1 + \tan^2 \theta \)

This identity is crucial because it connects the secant and tangent functions, allowing us to transform expressions involving \( \sec^2 \theta \) into expressions with \( \tan^2 \theta \). Understanding this identity helps to simplify the expression under the square root, which is a critical step in the solution.
Other identities may include those relating sine, cosine, and other trigonometric functions, which provide a foundation for manipulating and simplifying various trigonometric expressions. Remember, these identities are tools to transform and simplify calculations.

Simplifying Expressions

Simplifying expressions is the task of rewriting an expression in a shorter or more understandable form. In the process of solving this exercise, simplifying involved several key steps:

• Substituting \( x \) with \( a \sec \theta \) and simplifying the expression \( \sqrt{ (a \sec \theta)^2 - a^2 } \).
• Using the identity \( \sec^2 \theta = 1 + \tan^2 \theta \) to rewrite and evaluate expressions inside the square root.

After substituting \( a \sec \theta \), we calculated that:
\( (a \sec \theta)^2 - a^2 = a^2 (\sec^2 \theta - 1) = a^2 \tan^2 \theta \) which simplifies under the root as \( a \tan \theta \).
Finally, substituting back into the original equation, we derived \( \frac{\sin \theta}{a} \), showing how simplification turns a complex expression into a more manageable form. Each step requires attention to ensure correctness, and involves manipulating the equation to progressively simplify it.

Secant and Tangent Functions

Secant and tangent are two fundamental trigonometric functions typically dealing with right triangles and the unit circle. They are defined as:

• Secant (\(\sec \theta\)): the reciprocal of cosine. It is expressed as \( \sec \theta = \frac{1}{\cos \theta} \).
• Tangent (\(\tan \theta\)): the ratio of sine to cosine. It is written as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

In the context of this exercise, we employed these functions to express and simplify the expression given the substitution \( x = a \sec \theta \). The goal was to reduce the expression \( \sqrt{x^2 - a^2} \) into something involving tangents and simplify it further. Employing \(\sec\) and \(\tan\) in combination allows us to take advantage of their identities to ease the complexity of our expression. Overall, secant and tangent functions, along with their identities, are powerful tools in simplifying more intricate trigonometric problems.