Problem 93 - Algebra and Trigonometry

Question

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7 ) Assume \(0 \leq \theta<\pi / 2\) $$ \sqrt{9-x^{2}}, \quad x=3 \sin \theta $$

Step-by-Step Solution

Answer

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

Step 1: Substitute the Trigonometric Expression

Start by substituting \( x = 3 \sin \theta \) into the expression \( \sqrt{9-x^{2}} \). This gives us \( \sqrt{9 - (3 \sin \theta)^2} \).

Step 2: Simplify Inside the Square Root

Calculate \( (3 \sin \theta)^2 = 9 \sin^2 \theta \). Substitute this back into the expression to get \( \sqrt{9 - 9 \sin^2 \theta} \).

Step 3: Factor Out Common Terms

Factor out the common factor of 9 inside the square root: \( \sqrt{9(1 - \sin^2 \theta)} \).

Step 4: Use Trigonometric Identity

Use the identity \( 1 - \sin^2 \theta = \cos^2 \theta \) to further simplify \( \sqrt{9(\cos^2 \theta)} \) to \( \sqrt{9} \cos \theta \).

Step 5: Simplify the Expression

Simplify \( \sqrt{9} \cos \theta \) to \( 3 \cos \theta \) because \( \sqrt{9} = 3 \) and given \( \theta \) is in the first quadrant, \( \cos \theta \) is non-negative.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry that simplify complex expressions. They are formulas involving trigonometric functions, like sine and cosine, that are universally true. In this exercise, we use the identity \( 1 - \sin^2 \theta = \cos^2 \theta \) to simplify the expression.

This identity is derived from the Pythagorean identity, which states:

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

By rearranging, we obtain \( \cos^2 \theta = 1 - \sin^2 \theta \). This equation is particularly useful when dealing with expressions involving sine because it allows us to rewrite them in terms of cosine. Using these identities makes the analysis of trigonometric functions more manageable and helps in simplifying expressions efficiently.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations. In mathematics, simplifying these expressions is often necessary to make them more understandable and to reduce complexity when solving problems.

In the given exercise, you start with the expression \( \sqrt{9 - x^2} \). By substituting the variable \( x \) with a trigonometric function, specifically \( x = 3 \sin \theta \), the expression becomes \( \sqrt{9 - (3\sin \theta)^2} \). This transformation is crucial as it allows us to leverage trigonometric identities for simplification. Recognizing how algebra and trigonometry can interact is key to mastering which type of substitution will provide the simplest result. This is a common strategy in calculus and other forms of higher mathematics.

Trigonometric Simplification

The process of trigonometric simplification involves breaking down complex trigonometric expressions into simpler forms. This often involves using identities or factoring to reduce the number of terms.

In this problem, after substituting and simplifying inside the square root, you get \( \sqrt{9(1 - \sin^2 \theta)} \). Utilizing the identity \( 1 - \sin^2 \theta = \cos^2 \theta \), the expression simplifies to \( \sqrt{9 \cos^2 \theta} \).

The steps of simplification involved:

• Substitution of the variable \( x \) into a trigonometric function.
• Using algebra to factor out constants from square roots.
• Applying trigonometric identities to finalize the expression.

This reveals the result: \( 3 \cos \theta \), a much simpler expression that is easier to evaluate or integrate. Trigonometric simplification thus aids in reducing the complexity of mathematical problems.

First Quadrant Angles

First quadrant angles are angles measured from 0 to \( \frac{\pi}{2} \), or 0 to 90 degrees in degrees measure. This range is significant because trigonometric functions in the first quadrant are typically straightforward:

• The sine of an angle is positive.
• The cosine of an angle is positive.
• The tangent of an angle (ratio of sine to cosine) is also positive.

When solving problems that involve these angles, especially those with substitutions like \( x = 3 \sin \theta \), knowing the sine and cosine will always yield positive results simplifies calculations. This ensures we handle signs correctly without needing extra adjustments.

In the given task, because we assume \( 0 \leq \theta < \pi / 2 \), \( \cos \theta \) is non-negative. Thus, simplifying \( \sqrt{9} \cos \theta \) leads directly to \( 3 \cos \theta \), safely assuming a non-negative outcome. Working in the first quadrant allows for easier problem-solving as it simplifies the process of predicting the sign and behavior of trigonometric outcomes. This understanding is critical when applying trigonometric concepts in calculus and other advanced math subjects.