Problem 90 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the substitution

We are given the expression \( \sqrt{1+x^2} \) and the substitution \( x = \tan \theta \). Begin by replacing \( x \) with \( \tan \theta \) in the expression.

Step 2: Substitute and simplify under the square root

Substitute \( x = \tan \theta \) into the expression \( \sqrt{1 + x^2} \). This becomes \( \sqrt{1 + (\tan \theta)^2} \). Use the trigonometric identity \( 1 + \tan^2 \theta = \sec^2 \theta \) to rewrite the expression as \( \sqrt{\sec^2 \theta} \).

Step 3: Further simplify the square root

Since \( \sqrt{\sec^2 \theta} = \sec \theta \) and given that \( 0 \leq \theta < \pi/2 \), \( \sec \theta \) is non-negative. Thus, we can simplify this to \( \sec \theta \).

Key Concepts

Understanding Trigonometric Identities

In trigonometry, identities are equations that hold true for all values of the involved variables. They are fundamental tools you use to simplify expressions and solve equations in algebra and calculus. A popular identity, used in our given exercise, is the Pythagorean identity: \( 1 + \tan^2 \theta = \sec^2 \theta \). This identity is derived from the Pythagorean theorem, and it plays a crucial role in the simplification of expressions involving squares of trigonometric functions.
When given \( \sqrt{1 + x^2} \) with \(x = \tan \theta\), substituting \(x\) leads to \( \sqrt{1 + \tan^2 \theta} \). By using the identity, this can be simplified to \( \sqrt{\sec^2 \theta} \), which directly simplifies to \( \sec \theta \).
Understanding and correctly applying trigonometric identities are key skills that help in maneuvering through complex trigonometric expressions efficiently.

Role of Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are not just ratios formed by sides of right-angled triangles; they have broader applications in various algebraic transformations as well. In the context of substitutions like the one given in this problem, the tangent function helps bridge an algebraic expression to a simpler trigonometric form.
Consider the substitution where \( x = \tan \theta \). This step transforms the algebraic \( x^2 \) into a form that allows us to use trigonometric identities, reducing the complexity. The secant function, which is \( \sec \theta = \frac{1}{\cos \theta} \), simplifies the solution further once the trigonometric identity is applied and the expression is simplified.
Hence, understanding how these functions convert algebraic expressions into familiar trigonometric entities is immensely useful in simplifying and solving mathematical problems.

Working with Algebraic Expressions

Algebraic expressions often involve variables and constants combined through operations like addition, subtraction, multiplication, and division. The challenge sometimes lies in transforming these expressions into more solvable forms, especially when trigonometric substitution is involved.
In the problem presented, \( \sqrt{1 + x^2} \) is an algebraic expression that is transformed using the substitution \( x = \tan \theta \). This substitution is strategic; it allows the expression to be rewritten in terms of trigonometric identities, which are often more familiar and more straightforward to work with.
The goal is always simplification: taking complex forms and rewriting them as something simpler, more recognizable, or easier to compute. Mastering this transformation process in algebraic contexts broadens your mathematical toolkit, enabling you to tackle a wide range of problems more efficiently.

• Identify potential substitutions that simplify the expression.
• Leverage known identities to rewrite and simplify.
• Reapply basic algebraic operations to achieve the simplest form.