Problem 65 - Precalculus: Functions and Graphs

Question

Make the trigonometric substitution \(x=a \sin \theta\) for \(-\pi / 2<\theta<\pi / 2\) and \(a>0 .\) Use fundamental identities to simplify the resulting expression. $$\left(a^{2}-x^{2}\right)^{3 / 2}$$

Step-by-Step Solution

Answer

The simplified expression is \(a^3 \cos^3 \theta\).

Step 1: Substitution

We begin by substituting \( x = a \sin \theta \) into the expression \((a^2 - x^2)^{3/2}\). This gives us \( x^2 = a^2 \sin^2 \theta\).

Step 2: Simplify Inside the Expression

Substitute \( x^2 = a^2 \sin^2 \theta\) into the expression: \[a^2 - x^2 = a^2 - a^2 \sin^2 \theta = a^2 (1 - \sin^2 \theta)\].

Step 3: Use Pythagorean Identity

Utilize the Pythagorean identity \(1 - \sin^2 \theta = \cos^2 \theta\) to further simplify: \[a^2 (1 - \sin^2 \theta) = a^2 \cos^2 \theta\].

Step 4: Expression Cubed

Now, take the expression inside the parentheses \[a^2 \cos^2 \theta\] and raise it to the power of \(\frac{3}{2}\):\[(a^2 \cos^2 \theta)^{3/2} = (a^2)^{3/2} (\cos^2 \theta)^{3/2}\].

Step 5: Final Simplification

Simplify \((a^2)^{3/2}\) as \(a^3\) since \((a^2)^{3/2} = a^3\), and \((\cos^2 \theta)^{3/2}\) as \(\cos^3 \theta\): \[(a^2 \cos^2 \theta)^{3/2} = a^3 \cos^3 \theta\].

Key Concepts

Pythagorean Identities

Pythagorean identities are vital tools in trigonometry, helping to simplify expressions and solve equations involving trigonometric functions. These identities are derived from the Pythagorean theorem, which relates the sides of a right triangle. One of the most commonly used Pythagorean identities is:

• \(1 = \sin^2 \theta + \cos^2 \theta\)

This identity helps in rewriting and simplifying expressions that involve squares of sine and cosine functions. In the context of our exercise, we specifically used:

• \(1 - \sin^2 \theta = \cos^2 \theta\)

This form of the identity is useful for transforming a subtraction of squared sine from one, into a squared cosine term, which can significantly simplify the algebra involved. Using these identities can turn complex trigonometric problems into more manageable forms. Remember that these identities apply to all angles, making them universally useful in precalculus and calculus applications.
Understanding and recognizing when to use Pythagorean identities is crucial in solving trigonometric problems efficiently. With practice, you'll find these transformations become second nature.

Simplifying Expressions

Simplifying expressions often involves breaking down complex terms into more elementary forms. This process is crucial for managing and solving mathematical problems, especially those in trigonometry.
In our example, we began with an expression involving a square root and trigonometric functions: \[(a^2 - x^2)^{3/2}\]
By substituting \(x = a \sin \theta\), the expression became \[a^2 (1 - \sin^2 \theta)\] which was further simplified using the Pythagorean identity into \[a^2 \cos^2 \theta\]. Next, this expression was simplified by raising it to the power of \(\frac{3}{2}\):

• \((a^2 \cos^2 \theta)^{3/2} = (a^2)^{3/2} (\cos^2 \theta)^{3/2}\)
• Simplifying further, \((a^2)^{3/2} = a^3\) and \((\cos^2 \theta)^{3/2} = \cos^3 \theta\)

It’s important to approach the simplification step-by-step:

• Make strategic substitutions to reduce complexity.
• Apply identities to transform expressions.
• Carefully perform algebraic operations to reach a simpler result.

Simplifying helps not only in minimizing the number of terms but also in making the problem easier to interpret and solve. It’s a key skill in both precalculus and calculus.

Precalculus Mathematics

Precalculus mathematics forms the foundation for calculus and includes an array of topics like algebra, geometry, and trigonometry. It is in this realm that trigonometric substitutions and identities play crucial roles.
Precalculus equips students with the tools to handle calculus concepts by exploring functions, complex numbers, and other mathematical structures. Trigonometric substitution, for instance, is a significant technique to learn. It involves replacing a variable with a trig function to simplify integration or solve equations.
This step-by-step substitution allows us to transform a complicated algebraic problem into an easier trigonometric form. Trigonometric functions bring in relationships based on right triangles, making them uniquely useful in applications across many fields of mathematics.

• Prepares students to deal with calculus problems more effectively by providing them with strategies and tools.
• Helps with understanding the geometric interpretation of algebraic expressions.

Practice in precalculus mathematics ensures a solid grasp of these fundamental concepts which are indispensable for higher-level math studies. It’s about mastering the types of transformations needed for future exploration in calculus and beyond.