Problem 38 - Algebra 2

Question

For Exercises \(38-40,\) consider \(f(x)=\sin ^{-1} x+\cos ^{-1} x\) Make a table of values, recording \(x\) and \(f(x)\) for \(x=\left\\{0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1,-\frac{1}{2},\right.\) \(-\frac{\sqrt{2}}{2},-\frac{\sqrt{3}}{2},-1 \\}\)

Step-by-Step Solution

Answer

For all given \( x \), \( f(x) = \frac{\pi}{2} \).

Step 1: Understanding the Function

We need to evaluate the function \( f(x) = \sin^{-1}x + \cos^{-1}x \) for different values of \( x \). It is known that \( \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \) for all \( x \) such that \( -1 \leq x \leq 1 \).

Step 2: Creating the List of Values for \( x \)

Write down the values of \( x \) for which we need to find \( f(x) \), which are: \( 0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1, -\frac{1}{2}, -\frac{\sqrt{2}}{2}, -\frac{\sqrt{3}}{2}, -1 \).

Step 3: Calculating \( f(x) \) for Each Value of \( x \)

Since \( f(x) = \frac{\pi}{2} \) for all \( x \) in the specified range, calculate \( f(x) \) for each specific \( x \): - For \( x = 0 \), \( f(0) = \frac{\pi}{2} \)- For \( x = \frac{1}{2} \), \( f\left(\frac{1}{2}\right) = \frac{\pi}{2} \)- For \( x = \frac{\sqrt{2}}{2} \), \( f\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{2} \)- For \( x = \frac{\sqrt{3}}{2} \), \( f\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{2} \)- For \( x = 1 \), \( f(1) = \frac{\pi}{2} \)- For \( x = -\frac{1}{2} \), \( f(-\frac{1}{2}) = \frac{\pi}{2} \)- For \( x = -\frac{\sqrt{2}}{2} \), \( f(-\frac{\sqrt{2}}{2}) = \frac{\pi}{2} \)- For \( x = -\frac{\sqrt{3}}{2} \), \( f(-\frac{\sqrt{3}}{2}) = \frac{\pi}{2} \)- For \( x = -1 \), \( f(-1) = \frac{\pi}{2} \)

Step 4: Compile the Table of Values

Formulate a table using the values of \( x \) and verify consistently that \( f(x) = \frac{\pi}{2} \) for all given \( x \):| \( x \) | \( f(x) \) ||-----------|------------|| 0 | \(\frac{\pi}{2}\) || \(\frac{1}{2}\)| \(\frac{\pi}{2}\) || \(\frac{\sqrt{2}}{2}\) | \(\frac{\pi}{2}\) || \(\frac{\sqrt{3}}{2}\) | \(\frac{\pi}{2}\) || 1 | \(\frac{\pi}{2}\) || \(-\frac{1}{2}\) | \(\frac{\pi}{2}\) || \(-\frac{\sqrt{2}}{2}\) | \(\frac{\pi}{2}\) || \(-\frac{\sqrt{3}}{2}\) | \(\frac{\pi}{2}\) || -1 | \(\frac{\pi}{2}\) |

Key Concepts

Sine and Cosine Functions

Sine and cosine are two fundamental trigonometric functions that arise in the study of triangles and are pivotal in explaining periodic phenomena. The sine function, denoted as \( \sin \), measures the vertical component, or the length of the opposite side over the hypotenuse in a right triangle, at a given angle. Similarly, the cosine function \( \cos \) measures the adjacent side over the hypotenuse.Both sine and cosine are defined for angles typically measured in either degrees or radians. They fluctuate between -1 and 1, cycling in a sinusoidal pattern as the angle increases. Understanding these functions is crucial for solving problems in trigonometry, physics, and engineering.

Function Values and Tables

Evaluating and understanding the function \( f(x) = \sin^{-1}(x) + \cos^{-1}(x) \) is made more straightforward by using a table of values. Trigonometric functions often entail repetition over certain intervals, known as periodicity. Thus, knowing how to interpret and form such tables is vital.For the inverse sine \( \sin^{-1} \) and inverse cosine \( \cos^{-1} \), these functions express angles in radians from their respective sine and cosine values within the domain of [-1, 1]. Notably, for any given \( x \) in this domain, \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \), indicating a fixed sum relationship across the interval. Creating a table by plugging in different \( x \) values reveals the unwavering value of \( f(x) \), underscoring this remarkable property.

Properties of Trigonometric Functions

The study of trigonometric functions reveals many intriguing properties. One notable property of the inverse functions \( \sin^{-1} \) and \( \cos^{-1} \) is that their values sum to a constant for their mutual domain \([-1, 1]\). This is due to their complementary nature in evaluating an angle's sine and cosine, respectively.Focus on these key properties of trigonometric functions:

• Periodicity: Both sine and cosine functions repeat values in cycles, the foundation of their predictable behaviors.
• Symmetry: Sine is an odd function \( \sin(-x) = -\sin(x) \) and cosine is an even function \( \cos(-x) = \cos(x) \).
• Amplitude: The range from -1 to 1 indicates their amplitude limitations and is crucial in wave analysis.

Understanding these properties enhances problem-solving skills in various advanced mathematical scenarios.