Problem 92 - Algebra and Trigonometry

Question

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7\()\) . Assume that \(0 \leq \theta<\pi / 2 .\) $$ \sqrt{1+x^{2}}, \quad x=\tan \theta $$

Step-by-Step Solution

Answer

The simplified expression is \( \sec \theta \).

Step 1: Substitute the Expression

We are given the expression \( \sqrt{1 + x^2} \) with the substitution \( x = \tan \theta \). First, replace \( x \) with \( \tan \theta \). Our expression becomes \( \sqrt{1 + (\tan \theta)^2} \).

Step 2: Simplify Using Trigonometric Identity

Recall the trigonometric identity: \( 1 + \tan^2 \theta = \sec^2 \theta \). Substitute this identity into the expression to get \( \sqrt{\sec^2 \theta} \).

Step 3: Simplify the Expression

Simplify \( \sqrt{\sec^2 \theta} \). Since the square root and the square cancel each other out, it simplifies to \( \sec \theta \).

Step 4: Express the Result in Terms of \( \theta \)

The simplified expression is \( \sec \theta \), which is in terms of \( \theta \). Since \( \sec \theta = \frac{1}{\cos \theta} \), this can also be represented as \( \frac{1}{\cos \theta} \) if further simplification or conversion is needed.

Key Concepts

Algebraic Expression

An algebraic expression in mathematics is a combination of numbers, variables, and operational symbols. Variables are letters like \( x \) or \( \theta \) used to represent unknown or variable quantities. Algebraic expressions can be simple, like \( x + 5 \), or more complex, like \( \sqrt{1 + x^2} \).

• The given task involves an algebraic expression \( \sqrt{1 + x^2} \).
• We are instructed to make a substitution, where \( x \) is replaced with the trigonometric function \( \tan \theta \).
• This substitution transforms the algebraic expression into one involving trigonometric terms.

This process illustrates how algebra can seamlessly incorporate trigonometric functions, creating a bridge between algebra and trigonometry. It also exemplifies how expressions can be crafted to explore different mathematical perspectives and solutions.

Trigonometric Identities

Trigonometric identities are fundamental equations that hold true for all angles. They provide crucial relationships between different trigonometric functions like sine, cosine, and tangent.

• A commonly used trigonometric identity is \( 1 + \tan^2 \theta = \sec^2 \theta \).
• This identity is useful in trigonometric substitution problems, where algebraic expressions are transformed using trigonometric functions.
• In the given problem, once we substitute \( x = \tan \theta \) into \( \sqrt{1 + x^2} \), the identity \( 1 + \tan^2 \theta = \sec^2 \theta \) allows us to simplify the expression.

These identities simplify calculations and solutions, making them indispensable for solving equations involving trigonometric terms. They guide the shaping of expressions from algebraic to trigonometric, facilitating easier manipulation and understanding.

Simplification in Trigonometry

Simplification is a process to make expressions or equations easier to manage and understand. In trigonometry, simplification often involves transforming complex trigonometric expressions into more straightforward or more familiar forms.

• After substituting \( x = \tan \theta \), the expression \( \sqrt{1 + (\tan \theta)^2} \) is obtained.
• The identity \( 1 + \tan^2 \theta = \sec^2 \theta \) helps simplify this to \( \sqrt{\sec^2 \theta} \).
• The expression \( \sqrt{\sec^2 \theta} \) simplifies to \( \sec \theta \) since the square root and square cancel each other out.
• \( \sec \theta \) can also be expressed as \( \frac{1}{\cos \theta} \) for further simplification.

This process exemplifies how trigonometric expressions can be broken down using identities and basic arithmetic, enhancing both understanding and functionality in mathematical problem-solving.