Problem 95 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Substitute the given trigonometric identity

We start by substituting the trigonometric identity provided into the function. Given \( x = 3 \sin\theta \), substitute this into the expression \( \sqrt{9 - x^2} \). Thus, we have: \[ \sqrt{9 - (3 \sin \theta)^2} \]

Step 2: Simplify the expression

Simplify the expression: \[ \sqrt{9 - 9 \sin^2 \theta} \] Factor out the \( 9 \) from under the square root: \[ \sqrt{9(1 - \sin^2 \theta)} \] This follows as the term becomes: \[ \sqrt{9} \cdot \sqrt{1 - \sin^2 \theta} \]

Step 3: Apply trigonometric identity

Use the identity \( 1 - \sin^2 \theta = \cos^2 \theta \) to further simplify the expression: \[ \sqrt{9} \cdot \sqrt{\cos^2 \theta} \] Thus, it simplifies to: \[ 3 \cdot |\cos \theta| \]

Step 4: Evaluate absolute value under given conditions

Since it is given that \( 0 \leq \theta < \frac{\pi}{2} \), \( \cos \theta \) is non-negative. Hence, the absolute value can be removed and we are left with: \[ 3 \cos \theta \]

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships between the angles and sides of a triangle using trigonometric functions. They help us simplify expressions and solve various mathematical problems with ease.
Understanding these identities allows for the creative transformation of mathematical expressions, enabling better analysis and solution processes.One of the most widely used trigonometric identities is the Pythagorean identity:

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

This identity comes handy in simplifying expressions, just as we did with our original step-by-step solution. We substituted \( 1 - \sin^2 \theta \) with \( \cos^2 \theta \) to transform our problem into a more workable form.
This approach underscores the indispensable nature of trigonometric identities in mathematical problem-solving.

Trigonometric Simplification

Trigonometric simplification is the process of transforming trigonometric expressions into simpler forms to make calculations easier and more manageable. Often, simplifying expressions requires the clever use of trigonometric identities.In our exercise, we simplified \( \sqrt{9 - 9 \sin^2 \theta} \) by factoring out \( 9 \), resulting in \( \sqrt{9} \cdot \sqrt{1 - \sin^2 \theta} \). The key here was applying the identity we discussed earlier:

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

Through this simplification process, our expression became \( 3 \cdot |\cos \theta| \).
Understanding how to apply identities effectively, and recognize opportunities to simplify, is invaluable in solving mathematical problems.

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It connects the angle of a right triangle to the ratio of the adjacent side to the hypotenuse.In our exercise, understanding the cosine function was critical. We used the identity \( 1 - \sin^2 \theta = \cos^2 \theta \) to simplify our expression to \( 3 \cdot |\cos \theta| \).
Given the exercise specifies \( 0 \leq \theta < \frac{\pi}{2} \), knowing that \( \cos \theta \) is non-negative ensures \(|\cos \theta| = \cos \theta\).Here are some crucial aspects of the cosine function:

• Range: The cosine value varies between -1 and 1.
• Behavior: It is positive in the first quadrant, which aligns with our angle conditions.

Grasping these concepts around the cosine function aids in understanding more complex trigonometric relationships and enhances problem-solving skills.