Problem 72 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 69-72: Make the trigonometric substitution $$ x=a \tan \theta \text { for }-\pi / 2<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ \left(x^{2}+a^{2}\right)^{3 / 2} $$

Step-by-Step Solution

Answer

\( (x^2 + a^2)^{3/2} \) simplifies to \( a^3 \sec^3 \theta \).

Step 1: Perform the Substitution

Start by substituting the given trigonometric identity. Replace \( x \) in the expression \( \left(x^2 + a^2\right)^{3/2} \) with \( a \tan \theta \). Thus, the expression becomes: \[ \left( (a \tan \theta)^2 + a^2 \right)^{3/2}.\]

Step 2: Simplify the Inside Expression

Simplify inside the parentheses first. Expand \( (a \tan \theta)^2 \) to get \( a^2 \tan^2 \theta \). Therefore, the expression inside the parentheses is:\[ a^2 \tan^2 \theta + a^2.\] Factor out \( a^2 \):\[ a^2(\tan^2 \theta + 1).\]

Step 3: Use Trigonometric Identity

Recognize that \( \tan^2 \theta + 1 = \sec^2 \theta \), a known trigonometric identity. Substitute this into the expression to get:\[ a^2 \sec^2 \theta.\]

Step 4: Apply the Exponent

Now, raise the entire expression \( \left(a^2 \sec^2 \theta\right) \) to the power of \( 3/2 \): \[ \left(a^2 \sec^2 \theta \right)^{3/2}.\] Separate the terms as follows:\[ (a^2)^{3/2} (\sec^2 \theta)^{3/2}.\]

Step 5: Simplify Each Factor

Simplify each part:1. \( (a^2)^{3/2} = a^3 \).2. \( (\sec^2 \theta)^{3/2} = \sec^3 \theta \).Hence, the expression becomes:\[ a^3 \sec^3 \theta.\]

Step 6: Conclusion: Simplified Expression

The expression \( \left(x^2 + a^2\right)^{3/2} \) simplifies to \( a^3 \sec^3 \theta \) after performing the trigonometric substitution.

Key Concepts

Trigonometric Identities

When studying trigonometry, there are essential identities that we frequently use to simplify expressions. One crucial trigonometric identity is the Pythagorean identity related to tangent and secant:

• The identity \( \tan^2 \theta + 1 = \sec^2 \theta \) helps in transforming complex trigonometric expressions into simpler forms.
• This identity stems from the relationship between the sides of a right triangle, where the hypotenuse squared is equal to the sum of the squares of the other two sides.

Recognizing these identities can significantly aid in simplifying complex mathematical expressions. By substituting such identities, calculations become more manageable and the expressions become more straightforward to handle.

Simplifying Expressions

Simplifying expressions is a fundamental skill that not only involves algebraic manipulation but often integrates trigonometric identities. It's an essential step in solving equations or evaluating expressions.

• To begin simplifying expressions, identify and substitute any trigonometric identities that match parts of the expression.
• Once identified, replace the more complex parts with their simplified equivalents using known identities, like replacing \( \tan^2 \theta + 1 \) with \( \sec^2 \theta \).
• Further simplification involves factoring or expanding terms where needed and then applying exponents to simplify the entire expression further.

Let's apply these principles: by recognizing that \((a \tan \theta)^2 + a^2\) can be transformed using \(\tan^2 \theta + 1 = \sec^2 \theta\), it becomes \(a^2 \sec^2 \theta\). This is much simpler to work with in subsequent steps.

Secant Function

The secant function, denoted \(\sec \theta\), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function:\[\sec \theta = \frac{1}{\cos \theta}\]

• This function is particularly useful in handling expressions involving division by the cosine function, and it's commonly encountered in problems involving right triangles.
• In the context of trigonometric substitution, secant aids in the further simplification of expressions, like transforming \(\tan^2 \theta + 1\) using \(\sec^2 \theta\).

When substituting trigonometric expressions, leveraging \(\sec \theta\) can simplify the evaluation, as seen in \(a^3 \sec^3 \theta\). Overall, understanding and utilizing the secant function effectively allows for a smoother simplification process in calculus and trigonometry problems.