Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 23-30: Write the expression as an algebraic expression in \(x\) for \(x>0\). $$ \left(\sin ^{-1} \frac{x}{\sqrt{x^{2}+4}}\right) $$

Step-by-Step Solution

Answer

\( \tan^{-1} \frac{x}{2} \).

Step 1: Understanding the Inverse Trigonometric Function

We start with the expression \( \sin^{-1}\left( \frac{x}{\sqrt{x^2 + 4}} \right) \). This is an inverse sine expression, asking for the angle \( \theta \) whose sine is \( \frac{x}{\sqrt{x^2+4}} \).

Step 2: Set Up the Basic Trigonometric Identity

Since \( \theta = \sin^{-1} y\), then by definition \( \sin(\theta) = y\). Substitute the given expression: \( \sin(\theta) = \frac{x}{\sqrt{x^2+4}} \).

Step 3: Find the Cosine Using the Pythagorean Identity

Use the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) to obtain \( \cos(\theta) \). We have \( \sin^2(\theta) = \frac{x^2}{x^2 + 4} \). Therefore, calculate \( \cos^2(\theta) = 1 - \frac{x^2}{x^2 + 4} = \frac{4}{x^2 + 4} \).

Step 4: Calculate Cosine from the Identity

Since \( \cos^2(\theta) = \frac{4}{x^2 + 4} \), take the square root to find \( \cos(\theta) \), giving \( \cos(\theta) = \frac{2}{\sqrt{x^2 + 4}} \), considering \( x > 0 \).

Step 5: Simplifying Expression

The trigonometric identity for a right triangle with angle \( \theta \) gives \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{x/\sqrt{x^2+4}}{2/\sqrt{x^2+4}} = \frac{x}{2} \). Thus, the angle \( \theta = \tan^{-1} \frac{x}{2} \).

Step 6: Conclusion

The expression \( \sin^{-1} \frac{x}{\sqrt{x^{2}+4}} \) reduces to the angle: \( \tan^{-1} \frac{x}{2} \).

Key Concepts

Understanding Sin Inverse

In trigonometry, the sin inverse function, also known as arcsine, is crucial. It helps identify the angle whose sine value is a given number. The notation used is \( \sin^{-1}(x) \), and it applies when you know the sine of an angle and need the angle itself. This function outputs an angle in radians between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). In our problem, we have \( \sin^{-1}\left( \frac{x}{\sqrt{x^2 + 4}} \right) \). Here, \(x\) is the unknown variable, and the expression \(\frac{x}{\sqrt{x^2+4}}\) denotes the sine of the angle \(\theta\).

The Role of Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides are defined. They are essential in determining relationships within angles and sides of triangles. The most common identities include:

• Sine and Cosine Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Tangent Identity: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

In this exercise, we used these identities to transition from sine to cosine, which helped evaluate \(\theta\) and eventually led us to a new expression involving tangent. Understanding these identities is key to manipulating and simplifying trigonometric expressions.

Manipulating Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations. Manipulating these expressions is crucial to simplify or transform them. In this exercise, the expression to be simplified is \(\sin^{-1}\left(\frac{x}{\sqrt{x^2+4}}\right)\). Here are the crucial steps utilized:

• First, we wrote \(\frac{x}{\sqrt{x^2+4}}\) as the sine of an angle \(\theta\).
• Then applied the Pythagorean Identity to find \(\cos(\theta)\), simplifying it to \(\frac{2}{\sqrt{x^2+4}}\).
• Finally, using the tangent identity, we found that \(\tan(\theta) = \frac{x}{2}\), thus simplifying the original expression into an algebraic form.

The systematic simplification process highlights the power of algebra in solving complex problems.

Exploring The Pythagorean Identity

The Pythagorean identity in trigonometry is \(\sin^2(\theta) + \cos^2(\theta) = 1\). It comes from the Pythagorean theorem applied to a unit circle.In this context, the identity helps to find one trigonometric function in terms of another. When \(\sin(\theta) = \frac{x}{\sqrt{x^2+4}}\), the identity allows us to express \(\cos(\theta)\) as:\[\cos^2(\theta) = 1 - \sin^2(\theta) \Rightarrow \cos^2(\theta) = 1 - \frac{x^2}{x^2 + 4} = \frac{4}{x^2 + 4}\]This step is essential as it provided the basis to derive \(\cos(\theta)\) as \(\frac{2}{\sqrt{x^2+4}}\) and use it to reformulate the expression to \(\tan^{-1}\left(\frac{x}{2}\right)\). The Pythagorean identity is a foundational tool in trigonometry, offering a straightforward way to relate different functions.