Problem 151 - Precalculus

Question

For the following exercises, find the function if \(\sin t=\frac{x}{x+1}\) $$ \cos \left(\sin ^{-1}\left(\frac{x}{x+1}\right)\right) $$

Step-by-Step Solution

Answer

\(\cos(\sin^{-1}(\frac{x}{x+1})) = \frac{\sqrt{2x + 1}}{x+1}\)

Step 1: Understand the Problem

We need to find the function for \( \cos(\sin^{-1}(\frac{x}{x+1})) \). This involves recognizing that we have an inverse trigonometric function followed by a cosine function.

Step 2: Define \( \theta \) for \( \sin(\theta) \)

Let's assume \( \sin(\theta) = \frac{x}{x+1} \), where \( \theta = \sin^{-1}(\frac{x}{x+1}) \). Our goal is to determine \( \cos(\theta) \).

Step 3: Use the Pythagorean Identity

Recall the identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \). Substitute \( \sin(\theta) = \frac{x}{x+1} \) into this identity.

Step 4: Compute \( \sin^2(\theta) \)

Square \( \sin(\theta) \):\[\sin^2(\theta) = \left(\frac{x}{x+1}\right)^2 = \frac{x^2}{(x+1)^2}.\]

Step 5: Solve for \( \cos^2(\theta) \)

Using the identity, solve for \( \cos^2(\theta) \):\[\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{x^2}{(x+1)^2}.\]

Step 6: Simplify \( \cos^2(\theta) \)

Simplify the expression:\[\cos^2(\theta) = \frac{(x+1)^2 - x^2}{(x+1)^2} = \frac{x^2 + 2x + 1 - x^2}{(x+1)^2} = \frac{2x + 1}{(x+1)^2}.\]

Step 7: Determine \( \cos(\theta) \)

Since \( \cos(\theta) \) must be positive (considering the range of \( \sin^{-1} \) is \([-\pi/2, \pi/2]\)), we take the positive square root:\[\cos(\theta) = \sqrt{\frac{2x + 1}{(x+1)^2}} = \frac{\sqrt{2x + 1}}{x+1}.\]

Step 8: Final Result

Thus, the function is:\[\cos(\sin^{-1}(\frac{x}{x+1})) = \frac{\sqrt{2x + 1}}{x+1}.\]

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental to understanding the relationships between the trigonometric functions such as sine, cosine, and tangent. These identities help simplify complex expressions and solve equations involving trigonometric functions. In this exercise, we encounter the use of the Pythagorean identity, which is one of the most important identities:

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

This identity expresses a fundamental property of right triangles and the unit circle. By understanding this relationship, we can deduce other properties such as \( \sin(\theta) \) and \( \cos(\theta) \), even when we start with a complicated angle or expression. For example, when given \( \sin(\theta) \), this identity allows us to find the corresponding cosine value, as seen in the exercise.
Understanding these identities is crucial because they serve as the foundation to manipulate and solve various trigonometric expressions. Keep practicing, and soon you'll find these become second nature!

Pythagorean Identity

The Pythagorean Identity is a cornerstone in trigonometry. It is based on the geometric principles derived from the Pythagorean theorem applied to the unit circle. For any angle \( \theta \), the Pythagorean identity is written as:

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

This identity is particularly useful when given either \( \sin(\theta) \) or \( \cos(\theta) \) and needing to find the other. For instance, in our solution, we start with \( \sin(\theta) = \frac{x}{x+1} \). To find \( \cos(\theta) \), we use this identity by substituting \( \sin^2(\theta) \) and solving for \( \cos^2(\theta) \) as shown:

• \( \cos^2(\theta) = 1 - \sin^2(\theta) \)
• \( \cos^2(\theta) = 1 - \frac{x^2}{(x+1)^2} \)

This approach turns complex inverse trigonometric expressions into manageable calculations, enabling us to eventually solve for \( \cos(\theta) \) and further analyze the problem. It's a powerful tool in both theoretical mathematics and practical applications.

Function Composition

Function composition involves taking the output of one function and using it as the input for another. In mathematics, this is a common technique used to simplify expressions and solve more complex functions. In this problem, we are dealing with the composition of the inverse sine function with the cosine function:

• \( \cos(\sin^{-1}(u)) \)

Here, the inner function \( \sin^{-1}(u) \) returns an angle \( \theta \) such that \( \sin(\theta) = u \). This angle is then fed into the outer function, \( \cos( \theta ) \), to find its cosine value.
This method of handling inverse trigonometric functions is crucial because it allows us to break down otherwise intricate expressions into simpler, actionable steps. By first simplifying \( \sin(\theta) = \frac{x}{x+1} \), then using the Pythagorean identity, we ultimately find \( \cos( \theta ) \), demonstrating how function composition is effectively employed in solving trigonometric problems. Understanding this concept enables one to navigate through mixed trigonometric functions with confidence.