Problem 91 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \( \tan \theta \).

Step 1: Introduction of Substitution

We begin with the given algebraic expression \( \sqrt{x^2 - 1} \) and the substitution \( x = \sec \theta \). This indicates we can use trigonometric identities to simplify the expression further.

Step 2: Substitute \( x = \sec \theta \)

Replace \( x \) in the expression with \( \sec \theta \). This gives us \( \sqrt{(\sec \theta)^2 - 1} \).

Step 3: Apply Trigonometric Identity

Recall the identity \( \sec^2 \theta = 1 + \tan^2 \theta \). We substitute \( \sec^2 \theta \) with \( 1 + \tan^2 \theta \) in our expression, getting \( \sqrt{\tan^2 \theta} \).

Step 4: Simplify the Expression

The square root of \( \tan^2 \theta \) is \( \tan \theta \), so the expression simplifies to \( \tan \theta \).

Step 5: Verify the Range of \( \theta \)

We assumed \( 0 \leq \theta < \frac{\pi}{2} \), which corresponds to the first quadrant where \( \tan \theta \geq 0 \). Therefore, \( \sqrt{\tan^2 \theta} = \tan \theta \) is correctly simplified without needing an absolute value.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They are crucial tools in mathematics, especially in trigonometry, as they allow us to simplify and transform expressions. One of the most common identities used in problems concerning trigonometric substitution is the Pythagorean identity:

• \( \sec^2 \theta = 1 + \tan^2 \theta \)

This identity helps in transforming expressions involving secants or tangents into simpler forms. In our exercise, by using this identity, we turned a complex algebraic expression into something more manageable, showing the power of these identities in simplifying intricate problems.

Algebraic Expression

An algebraic expression is a mathematical phrase involving numbers, variables, and operations. In this context, the given expression \( \sqrt{x^2 - 1} \) represents an algebraic expression that we wish to simplify using trigonometric substitution. Substitution is a method where we replace a variable with an expression that simplifies the problem.

When substituting \( x = \sec \theta \), our algebraic expression becomes trigonometric, which can then be simplified using trigonometric identities. This approach turns our problem-solving strategy into a streamlined process, replacing complex algebraic forms with simpler trigonometric terms.

Secant Function

The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function: \( \sec \theta = \frac{1}{\cos \theta} \). It is one of the trigonometric functions often used in calculus and trigonometry.

In trigonometric substitution, particularly when dealing with expressions like \( \sqrt{x^2 - 1} \), substituting \( x = \sec \theta \) is a common tactic. This substitution leverages the identity \( \sec^2 \theta = 1 + \tan^2 \theta \), thereby transforming algebraic expressions into manageable trigonometric ones. This conversion often simplifies the process, making it easier to move forward with the problem.

Tangent Function

The tangent function, represented by \( \tan \theta \), is defined as the ratio of the sine and cosine of an angle: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). It plays a significant role in trigonometric calculations because it relates the angles to the lengths of the opposite and adjacent sides in a right triangle.

• In trigonometric substitution exercises, solving \( \sqrt{\tan^2 \theta} \) simplifies directly to \( \tan \theta \) when \( 0 \leq \theta < \frac{\pi}{2} \).
• This range ensures that the tangent is non-negative, making this operation straightforward.

Consequently, using the tangent function helps not only to simplify but also to verify the correctness of trigonometric expressions, ensuring they fit within the appropriate constraints. Thus, understanding tangent function properties is crucial in solving algebraic expressions through trigonometric substitution.