Problem 90 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(\mathbf{0}<\boldsymbol{\theta}<\pi / 2\) $$\sqrt{\left(x^{2}-16\right)^{3}}, \quad x=4 \sec \theta$$

Step-by-Step Solution

Answer

The algebraic expression \( \sqrt{\left(x^{2}-16\right)^{3}} \), when \( x = 4 \sec \theta \), simplifies to \( 16 \tan^3 \theta \).

Step 1: Substitute \( x \) in the expression

The algebraic expression is \( \sqrt{\left(x^{2}-16\right)^{3}} \). Replace \( x \) in the expression with \( 4 \sec \theta \), which will give \( \sqrt{\left((4\sec \theta)^{2}-16\right)^{3}} \).

Step 2: Simplify the expression

Simplify the expression inside the square root: \( \sqrt{\left((4\sec \theta)^{2}-16\right)^{3}} \) becomes \( \sqrt{\left(16\sec^2 \theta -16\right)^{3}} \). Factor out the common factor of 16 and simplify further to get \( \sqrt{16^3 \left(\sec^2 \theta -1\right)^{3}} \).

Step 3: Apply the trigonometric identity

Replace \( \sec^2 \theta -1 \) with \( \tan^2 \theta \). This is true because of the trigonometric identity \( \sec^2 \theta = \tan^2 \theta + 1 \). The expression becomes \( \sqrt{16^3 \tan^2 \theta ^{3}} \).

Step 4: Simplify the expression

Simplify the chosen expression further to reach the final answer. The square root of \( 16^3 \tan^2 \theta ^{3} \) gives \( 16 \tan^3 \theta \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics, especially useful in simplifying complex expressions involving trigonometric functions. An identity is essentially an equation that holds true for all values of the variable within a certain range. One of the most important identities used in this problem is the Pythagorean identity:

• \( \sec^2 \theta = \tan^2 \theta + 1 \)

This identity helps replace the expression \( \sec^2 \theta - 1 \) with \( \tan^2 \theta \), simplifying many problems involving trigonometric substitutions.
Trigonometric identities are powerful because they enable transformations that can turn complex expressions into ones that are easier to interpret or reduce further. Mastering these identities is crucial for solving trigonometric problems efficiently.

Algebraic Expression

Algebraic expressions can sometimes appear intimidating, especially when they include squares, cubes, or roots. In this exercise, we have the expression \( \sqrt{\left(x^{2}-16\right)^{3}} \).
Here, the variable substitution \( x = 4 \sec \theta \) simplifies the expression under the square root. This essential step turns a complex algebraic form into something manageable by introducing a trigonometric function.
Working with algebraic expressions often involves substitutions that connect algebraic terms to a familiar set of functions.
This process makes it easier to replace complex expressions with simpler ones using known identities, as is the case with \( \sec^2 \theta - 1 = \tan^2 \theta \).

Simplification Steps

Simplification steps in a mathematical problem are crucial for reaching an understandable and concise solution. They involve systematically reducing an expression without altering its value. In this exercise, the simplification stages were vital:

• First, substituting \( x \) with \( 4 \sec \theta \) simplified the structure of the underlying terms.
• Factoring out common factors like 16 further reduced the expression complexity.
• Applying the key trigonometric identity \( \sec^2 \theta - 1 = \tan^2 \theta \) was pivotal in transforming the expression to only involve \( \tan \theta \).

After these steps, the expression \( \sqrt{16^3 \tan^2 \theta ^{3}} \) was much simpler and direct computation led us to the clean and final form of \( 16 \tan^3 \theta \).
Learning these simplification steps equips you with a strategy to tackle similar problems more confidently.