Problem 81 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(\mathbf{0}<\boldsymbol{\theta}<\pi / 2\) $$\sqrt{x^{2}+9}, \quad x=3 \tan \theta$$

Step-by-Step Solution

Answer

After applying trigonometric substitution to the expression \( \sqrt{x^{2}+9}\), where \(x=3\tan(\theta)\), the final expression is \(3\sec\theta\).

Step 1: Apply Trigonometric Substitution

Firstly, we substitute \(x = 3\tan(\theta)\) into the given expression, which forms \( \sqrt{(3\tan(\theta))^{2}+9}\).

Step 2: Simplify the Expression using Trigonometric Identities

Simplify the expression using the basic trigonometric identity \(\tan^{2}\theta + 1 = \sec^{2}\theta\). This gives us \(\sqrt{9(\tan^{2}\theta +1)}\).

Step 3: Further Simplify the Expression

To further simplify, we recognize that inside the square root, \(9(\tan^{2}\theta +1)\) is equal to \(9\sec^2\theta\). This way, \(\sqrt{x^{2}+9}\) becomes \(\sqrt{9\sec^2\theta}\).

Step 4: Final Simplification

We simplify further by taking the square root, which gives us \(3\sec\theta\).

Key Concepts

Understanding Algebraic Expressions

An algebraic expression contains numbers, variables, and operations such as addition, multiplication, or trigonometric operations. For the given exercise, the algebraic expression is \( \sqrt{x^{2}+9} \), which is defined by combining a variable \( x \) with constants to create a mathematical phrase.
It is crucial to understand algebraic expressions because they form the basis of algebra and calculus. Manipulating these expressions often involves applying substitutions or transformations to explore different aspects of the problem.
In this problem, a trigonometric substitution \( x = 3 \tan(\theta) \) is used to express the algebraic expression in a more simplified form. This step simplifies the operation that initially might look complex, by relating it to a trigonometric function of \( \theta \).
Applying trigonometric substitution can be handy in a range of problems, especially when dealing with square roots that resemble certain classical forms.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. They are vital in converting complex expressions into more manageable forms. In this particular exercise, we employ several identities, notably \( \tan^2\theta + 1 = \sec^2\theta \).
This identity is central because it allows us to turn a \( \tan \) function, which emerged from the trigonometric substitution, into a \( \sec \) function. When \( x = 3 \tan \theta \) was placed into the equation, the expression developed into \( \sqrt{(3\tan\theta)^2 + 9} \).
Recognizing the identity within the expression inside the square root simplifies the problem significantly and provides a pathway to formulating the final solution in terms of a single trigonometric function.

Simplification Process

Simplification helps in reducing mathematical expressions into simpler forms. It streamlines complex calculations and clarifies the relationships between parts of the problem.

• Begin with substituting \( x = 3 \tan \theta \) making the expression \( \sqrt{9 \tan^2\theta + 9} \).
• Apply the trigonometric identity \( \tan^2\theta + 1 = \sec^2\theta \) to simplify to \( \sqrt{9 \sec^2 \theta} \).
• Ultimately, this becomes \( 3 \sec \theta \) after taking the square root.

Through these steps, the originally daunting expression fades into a recognizable and simplified trigonometric expression, illustrating the power of basic algebra combined with trigonometric identities. It is about constantly seeking patterns that lead towards a more elegant expression.

Mastering Square Roots

A square root is a mathematical operation where you find a number that, when multiplied by itself, gives you the original number under the square root symbol. For example, considering \( \sqrt{9\sec^2\theta} \) leads to the simplification to \( 3\sec\theta \) since the square root of 9 is 3.
Square roots often appear in problems dealing with quadratic forms or geometric lengths. They require thoughtful management because squaring and taking roots affects the magnitude and units of expressions. With trigonometric substitution, the presence of square roots can be managed by reducing the expression neatly to involve a single trigonometric term that retains the properties of the original function.
Understanding how square roots transform the expressions can enhance your mathematical insight, easing the path from complex statements to elegant solutions.