Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(\mathbf{0}<\boldsymbol{\theta}<\pi / 2\) $$\sqrt{4-x^{2}}, \quad x=2 \cos \theta$$

Step-by-Step Solution

Answer

The algebraic expression \( \sqrt{4-x^2} \) with \( x = 2 \cos \theta \) is then equivalent to \( 2 \sin \theta \) in terms of trigonometric functions.

Step 1: Substituting the given variable

Substitute \( x = 2 \cos \theta \) into our algebraic equation, that will result in: \( \sqrt{4 - (2 \cos \theta)^2} \)

Step 2: Simplifying the expression

Now simplifying the square rooted expression results in: \( \sqrt{4 - 4 \cos^2 \theta} \). This can also be factored as: \( \sqrt{4(1 - \cos^2 \theta)} \)

Step 3: Applying Trigonometric Identity

Now the expression inside the square root can be rewritten using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \). So, \( 1 - \cos^2 \theta = \sin^2 \theta \). That changes our expression now to: \( \sqrt{4 \sin^2 \theta} \)

Step 4: Simplifying the square root

Simplify the square root to result in: \( 2 |\sin \theta| \)

Step 5: Removing absolute value

Given that \( 0 < \theta < \frac{\pi}{2} \), \( \sin \theta \) is positive over this range, so we can remove the absolute value bars, which results in: \( 2 \sin \theta \)

Key Concepts

Algebraic Expression to Trigonometric Function

When dealing with integrals or other mathematical problems, converting algebraic expressions into trigonometric functions can be incredibly useful. This technique, known as trigonometric substitution, simplifies expressions containing square roots or specific power functions by leveraging the properties of trig functions.

In our exercise example, the algebraic expression \( \sqrt{4 - x^2} \) can be transformed with the substitution \( x = 2 \cos \theta \). This choice is based on the knowledge that \( \cos^2 \theta \) and \( \sin^2 \theta \) add up to 1, according to the Pythagorean identity, a fundamental relationship in trigonometry. By making this substitution, we shift the problem into the trigonometric realm, where it's further manipulable through the trigonometric identities.

Pythagorean Identity

The Pythagorean identity is one of the core principles in trigonometry and is crucial in simplifying trigonometric expressions. It states that for any angle \( \theta \), \( \cos^2 \theta + \sin^2 \theta = 1 \). This identity originates from the Pythagorean theorem related to the sides of a right triangle.

The identity is not just an abstract equation; it plays a practical role in substitutions like the one in our example. Here's why: after substituting \( x \) with \( 2 \cos \theta \), we get \( 4 - 4\cos^2 \theta \). Applying the Pythagorean identity, we can replace \( 1 - \cos^2 \theta \) with \( \sin^2 \theta \) making it possible to simplify the square root further. This step shows how Pythagorean identity directly assists in converting and simplifying algebraic expressions into trigonometric terms.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a common and essential process in high-level math, and understanding how to do it correctly can save time and reduce errors. After employing the Pythagorean identity in our example, we arrive at the simplified form: \( \sqrt{4 \sin^2 \theta} \). The next step involves addressing the square root of a squared function, which simplifies to the absolute value of the function inside: \( 2 |\sin \theta| \).

However, because \( \sin \theta \) is known to be positive within the domain \( 0 < \theta < \frac{\pi}{2} \), we can confidently remove the absolute value sign leading to a final answer of \( 2 \sin \theta \). This part of the process not only simplifies the expression but also illustrates the importance of understanding the constraints and behavior of trigonometric functions within specific intervals.