Problem 94 - Algebra and Trigonometry

Question

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7 ) Assume \(0 \leq \theta<\pi / 2\) $$ \frac{\sqrt{x^{2}-25}}{x}, x=5 \sec \theta $$

Step-by-Step Solution

Answer

The expression simplifies to \( \sin \theta \).

Step 1: Understand the Substitution

We're given the expression \( \frac{\sqrt{x^2 - 25}}{x} \) and the substitution \( x = 5 \sec \theta \). We need to substitute \( x = 5 \sec \theta \) into the expression and simplify.

Step 2: Substitute x in the Expression

Substitute \( x = 5 \sec \theta \) into the expression: \[ \frac{\sqrt{(5 \sec \theta)^2 - 25}}{5 \sec \theta} \] This simplifies to \[ \frac{\sqrt{25 \sec^2 \theta - 25}}{5 \sec \theta} \]

Step 3: Simplify Inside the Square Root

Factor out 25 from the square root expression inside: \[ \sqrt{25(\sec^2 \theta - 1)} \] This simplifies to \[ 5 \sqrt{\sec^2 \theta - 1} \] using \( \sqrt{25} = 5 \).

Step 4: Use Trigonometric Identity

Recall the identity \( \sec^2 \theta - 1 = \tan^2 \theta \). Substitute this into the expression:\[ 5 \sqrt{\tan^2 \theta} \] which simplifies to \[ 5 \tan \theta \] since \( \sqrt{\tan^2 \theta} = \tan \theta \).

Step 5: Simplify the Entire Expression

Substitute back into the original fractional expression:\[ \frac{5 \tan \theta}{5 \sec \theta} \] This simplifies to \[ \frac{\tan \theta}{\sec \theta} \].

Step 6: Final Simplification

Use the trigonometric identities \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \) to simplify:\[ \frac{\frac{\sin \theta}{\cos \theta}}{\frac{1}{\cos \theta}} = \sin \theta \] Therefore, the simplified expression is \( \sin \theta \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in trigonometry. They relate various trigonometric functions with one another. In this exercise, the identity used is \( \sec^2 \theta - 1 = \tan^2 \theta \). This identity helps in simplifying expressions involving secant (\( \sec \theta \)) and tangent (\( \tan \theta \)). It's derived from the Pythagorean identity \( \sec^2 \theta = 1 + \tan^2 \theta \).
When we perform substitutions involving trigonometric functions, remembering these identities can dramatically streamline our calculations.

• \( \sec \theta \) stands for "secant theta," which is the reciprocal of cosine, meaning \( \sec \theta = \frac{1}{\cos \theta} \).
• \( \tan \theta \) represents "tangent theta," which is the ratio of sine over cosine, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

In solving the problem, knowing that \( \tan^2 \theta \) is equal to \( \sec^2 \theta - 1 \) simplifies the square root to \( \tan \theta \). This vital step reduces complexity and allows the expression to eventually simplify to \( \sin \theta \).

Simplification of Expressions

Simplification is the process of reducing a mathematical expression to its most basic form. In this context, we took the expression \( \frac{\sqrt{x^2 - 25}}{x} \) and transformed its complex form into a simpler one.
The initial step involves substituting \( x = 5 \sec \theta \), which transforms the original expression. From there, factorization and trigonometric identities guide us through the simplification process.Simplification often involves:

• Factoring common elements, like the 25 in \( 25(\sec^2 \theta - 1) \).
• Breaking down complex terms using trigonometric identities, reducing \( \sqrt{\sec^2 \theta - 1} \) to \( \tan \theta \).
• Canceling out terms wherever applicable, which in our case, left us with \( \frac{\tan \theta}{\sec \theta} \).

Ultimately, simplification makes the expression easier to understand and reduces the effort needed in further calculations, while maintaining mathematical accuracy.

Algebraic Expression

Algebraic expressions consist of numbers, variables, and operations used to structure a mathematical phrase. In this exercise, the algebraic expression \( \frac{\sqrt{x^2 - 25}}{x} \) contains a blend of variables, radicals, and division.
Trigonometric substitution shifts algebraic expressions into a form where they can be easier to manipulate using trigonometric functions. This technique is particularly useful when working with expressions involving sums or differences of squared terms. By substituting \( x = 5 \sec \theta \), the complexity of the radical \( \sqrt{x^2 - 25} \) is reduced significantly.Throughout this problem, we've demonstrated:

• How variable substitution can transform and simplify an algebraic structure.
• The power of trigonometric identities in reducing expressions to simpler forms.

Substituting trigonometric functions proved to be a powerful strategy, especially as it led to the straightforward result of \( \sin \theta \), simplifying the algebraic expression entirely.