Problem 95 - Precalculus with Limits

Question

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of \( \theta \), where \( 0 < \theta < \pi/2 \). \( \sqrt{16 - x^2} \), \( x = 4 \sin \theta \)

Step-by-Step Solution

Answer

The algebraic expression \( \sqrt{16 - x^2} \) can be written as a trigonometric function \(16 \cos \theta\).

Step 1: Substitution

Start by substituting \( x = 4 \sin \theta \) into the given algebraic expression. So, we have: \(\sqrt{16 - x^2} \rightarrow \sqrt{16 - (4 \sin \theta)^2}\).

Step 2: Simplify the Expression

Simplify the expression inside the square root: \( \sqrt{16 - 16 \sin^2 \theta}\).

Step 3: Common Factor

16 is a common factor, we can take it out from the square root: \(16 \sqrt{1 - \sin^2 \theta}\).

Step 4: Trigonometric Identity

Now, use the Pythagorean Identity \(\cos^2 \theta = 1 - \sin^2 \theta\). So, \(16 \sqrt{1 - \sin^2 \theta}\) becomes \(16 \cos \theta\).

Step 5: Final Expression

By performing the substitution and simplification, the algebraic expression \(\sqrt{16 - x^2}\) is finally written as \(16 \cos \theta\) in terms of trigonometric function of \( \theta \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental to solving complex mathematical expressions involving angles and lengths. They allow us to simplify and transform equations by replacing one trigonometric function with another equivalent function.

Here's how it works:

• **Basic Trigonometric Functions**: Sine (\(\sin\theta\)), Cosine (\(\cos\theta\)), and Tangent (\(\tan\theta\)) are the primary trigonometric functions.
• **Derived Functions**: Secant (\(\sec\theta\)), Cosecant (\(\csc\theta\)), and Cotangent (\(\cot\theta\)) are derived from the primary functions.
• **Identities** help us relate different trigonometric functions and can simplify complex trigonometric expressions.

For example, in the original exercise, the use of a trigonometric identity allows us to convert an algebraic expression into an easier-to-understand trigonometric function. This transformation makes it simpler to evaluate, especially when the variable is replaced with a specific trigonometric substitution.

Pythagorean Identity

The Pythagorean identity is one of the most important identities in trigonometry, and it derives from the Pythagorean Theorem. It states that for any angle \(\theta\), the following relation holds:

\[\sin^2 \theta + \cos^2 \theta = 1\]
This identity is a cornerstone because:

• It relates the squares of the sine and cosine functions directly to 1.
• It is extremely useful for simplifying trigonometric expressions, such as converting between \(\sin\) and \(\cos\).
• Recognizing this identity in problems allows us to substitute and simplify equations, reducing their complexity.

In the given exercise, this identity is used to transform an algebraic expression within a square root. Because \(\cos^2 \theta = 1 - \sin^2 \theta\), we substitute \(1 - \sin^2 \theta\) with \(\cos^2 \theta\). This results in a much simpler expression that directly involves the cosine function, making the whole equation easier to handle.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They form the basis for equations and are essential in expressing real-world phenomena mathematically. Understanding how to manipulate these expressions is crucial for solving complex problems.

Here’s what to remember about algebraic expressions:

• **Components**: They consist of constants, coefficients (numerical values multiplying the variables), variables, and arithmetic operations.
• **Simplification**: Simplifying algebraic expressions often involves combining like terms, factoring, and performing arithmetic operations.
• **Substitution**: Sometimes, you replace variables with numbers or other expressions, as seen in the original problem where \(x = 4 \sin \theta\).

In the exercise, the algebraic expression \(\sqrt{16 - x^2}\) is transformed using trigonometric substitution. This involves the substitution \(x = 4 \sin \theta\), which simplifies the expression by leveraging trigonometric identities. As a result, the expression transforms into a trigonometric form, which is often easier to solve or evaluate, especially within a specified domain like \(0 < \theta < \frac{\pi}{2}\).