Problem 93 - Precalculus with Limits

Question

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of \( \theta \), where \( 0 < \theta < \pi/2 \). \( \sqrt{9 - x^2} \), \( x = 3 \cos \theta \)

Step-by-Step Solution

Answer

The algebraic expression \( \sqrt{9 - x^2} \), rewritten as a trigonometric function of \( \theta \) by using the substitution \( x = 3 \cos \theta \), comes out to be \(3\sin \theta\).

Step 1: Substitute the given equation

Replace \( x \) with \( 3 \cos \theta \) in the equation \( \sqrt{9 - x^2} \). This gives us \( \sqrt{9 - (3 \cos \theta)^2} \).

Step 2: Simplify the equation

Expand the equation to get \( \sqrt{9 - 9\cos^2 \theta} \).

Step 3: Factor out constants from the radical

We can factor the number 9 from the radical which will give us \( \sqrt{9(1 - \cos^2 \theta)} = 3\sqrt{1 - \cos^2 \theta} \).

Step 4: Use Trigonometric Identity

Now, replace \(1 - \cos^2 \theta\) with \( \sin^2 \theta \) because based on the trigonometric identity \( \sin^2 \theta + \cos^2 \theta = 1 \), \(1 - \cos^2 \theta\) stands for \( \sin^2 \theta \). Thus, the equation becomes \(3\sqrt{\sin^2 \theta}\).

Step 5: Simplify the equation

The square root and the square cancel out, leaving the final result as \(3\sin \theta\).

Key Concepts

Trigonometric Identity

Understanding trigonometric identities is an essential part of solving many mathematical problems, especially when dealing with trigonometric substitution. One of the fundamental identities that students must familiarize themselves with is the Pythagorean identity, which states that for any angle \( \theta \), \[ \sin^2 \theta + \cos^2 \theta = 1 .\] This identity is crucial when converting algebraic expressions involving radicals to trigonometric functions, as it allows for the simplification of the expressions. In our exercise example, the given expression \( \sqrt{9 - x^2} \) turns into a trigonometric function by applying the Pythagorean identity. As the algebraic expression within the radical includes a square term \( x^2 \), you can think of \( 9 \) as the hypotenuse squared in a right triangle, and \( x^2 \) as the adjacent side squared.
Using the identity, we can relate \( 1 - \cos^2 \theta \) to \( \sin^2 \theta \), essentially exchanging one function for the other. It's like having different descriptions of the same picture - whether we talk in terms of the cosine squared or the sine squared, we're actually describing the same ratio, but from different angles (pun intended).

Algebraic Expression

An algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division, and exponentiation by an exponent that is a rational number). When you encounter an algebraic expression such as \( 9 - x^2 \), you can see it as a representation of a particular mathematical relationship. After the substitution of \( x \) with \( 3 \cos \theta \), the expression transforms into \( 9 - (3 \cos \theta)^2 \), illustrating how variables can be replaced by trigonometric functions to reveal a deeper connection between algebra and trigonometry.
This process opens up a pathway to simpler forms and, eventually, a solution through radical simplification. In fact, algebraic substitution is a bridge that allows you to transport an algebraic concept into the realm of trigonometry, giving you the tools to operate within that new domain. It is similar to changing the language of the problem to uncover a solution that may be more readily apparent in trigonometric terms rather than algebraic ones.

Radical Simplification

Radical simplification involves the process of taking a complex expression involving roots and reducing it to a more basic or more easily interpreted form. Specifically, it comes in very handy when you're dealing with square roots, as you often do in trigonometry. For instance, in the exercise, the expression inside the radical becomes \( 9(1 - \cos^2 \theta) \) after factoring out the common term. Simplifying this using the Pythagorean identity eventually gives you \( 3 \sqrt{\sin^2 \theta} \).
At this point, considering that the square and the square root are inverse operations, they cancel each other out. As a result, this simplification leads us to \( 3\sin \theta \) which is a much neater and more usable expression. Think of radical simplification like peeling layers off an onion - each layer represents a mathematical operation that you strip away to get to the core of the expression. It's a process that requires a solid grasp of both algebraic manipulation and understanding of trigonometric relationships.