Problem 96 - Precalculus with Limits

Question

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of \( \theta \), where \( 0 < \theta < \pi/2 \). \( \sqrt{49 - x^2} \), \( x = 7 \sin \theta \)

Step-by-Step Solution

Answer

The algebraic expression \( \sqrt{49 - x^2} \) can be written as a trigonometric function of \( \theta \) as \( 7 \cos \theta \) when \( x = 7 \sin \theta \).

Step 1: Apply the trigonometric substitution

Replace \( x \) in the expression \( \sqrt{49 - x^2} \) with \( 7 \sin \theta \).\n\nSo we get \( \sqrt{49 - (7 \sin \theta)^2} \)

Step 2: Simplify the expression

We now simplify the expression inside the square root to get:\n\n\( \sqrt{49 - 49 \sin^2 \theta} \)

Step 3: Utilize a trigonometric identity

Recognize that \( 49 - 49 \sin^2 \theta \) can be written as \( 49(1 - \sin^2 \theta) \). Then realize that \( 1 - \sin^2 \theta \) is \( \cos^2 \theta \). So we can rewrite the expression as \( \sqrt{49 \cos^2 \theta} \).

Step 4: Simplify the square root

Simplify the square root to find the final answer: \n\n7 \cos \theta

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in solving and simplifying expressions involving trigonometric functions. These identities relate different trigonometric functions to each other, allowing for substitutions and simplifications. One of the most fundamental identities is the Pythagorean identity, which states that for any angle \( \theta \):

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

This identity plays a crucial role in many trigonometric substitution problems and is used extensively to simplify expressions.
In the exercise provided, the expression \( 49(1 - \sin^2 \theta) \) could be simplified using the identity \( 1 - \sin^2 \theta = \cos^2 \theta \). Recognizing and applying these identities can help transform complex algebraic expressions into more manageable forms, such as when we reduced \( \sqrt{49 - 49 \sin^2 \theta} \) to \( \sqrt{49 \cos^2 \theta} \).

Algebraic Expressions

Understanding algebraic expressions is vital when converting between different mathematical forms. Algebraic expressions are combinations of numbers, variables, and arithmetic operations. Here, the task is to transform the expression \( \sqrt{49 - x^2} \) into a trigonometric function using the substitution \( x = 7 \sin \theta \).
First, substitute \( x \) in the expression with \( 7 \sin \theta \). This results in \( \sqrt{49 - (7 \sin \theta)^2} \). Simplifying inside the square root leads to \( \sqrt{49 - 49 \sin^2 \theta} \). Rewriting algebraic terms using known identities, as shown, is key in transforming and simplifying expressions effectively. Applying such techniques ensures expressions are rewritten in their simplest possible forms, which can then be solved or analyzed further.

• Algebraic expressions can sometimes look intimidating.
• Breaking them down into smaller components helps in understanding and simplifying them.
• Using substitutions and identities is a clever way to maneuver complex problems.

Simplifying Expressions

Simplifying expressions involves reducing them to an easier or more manageable form. This is a technique widely used in both algebra and trigonometry. Simplification often involves substituting variables, factoring, and applying identities. For instance, the expression \( \sqrt{49 - 49 \sin^2 \theta} \) was simplified step-by-step:

• Factor out the common term: \( 49(1 - \sin^2 \theta) \).
• Utilize the identity \( 1 - \sin^2 \theta \equiv \cos^2 \theta \).
• Simplify to \( \sqrt{49 \cos^2 \theta} = 7 \cos \theta \).

The final expression, \( 7 \cos \theta \), represents the simplified form of the original algebraic expression, rewritten as a trigonometric function. Simplification can involve multiple steps, but by breaking down the process, each step becomes clearer. Thus, students can improve their problem-solving skills and confidence with practice, leading to mastery in tackling similar expressions in the future.