Problem 86 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. \(y=\arcsin \frac{x}{2}\)

Step-by-Step Solution

Answer

The graph of the function \(y=\arcsin \frac{x}{2}\) is an increasing curve starting at (-2,-\(\frac{\pi}{2}\)) and ending at (2,\(\frac{\pi}{2}\)). Note that the function is increasing but at a slower rate compared to \(y=\arcsin x\). The x-values will range from -2 to 2 and y-values will range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Step 1: Understand the Characteristics of the Function

Recognize that \(\arcsin x\) (also known as 'inverse of sine function') is defined for -1 ≤ x ≤ 1 and the output (y-values) of the function is in the interval \(-\frac{\pi}{2}\leq y \leq \frac{\pi}{2}\). This function is increasing in its domain.

Step 2: Adapt Characteristics for Given Function

The given function \(y=\arcsin \frac{x}{2}\) will have a domain for -2 ≤ x ≤ 2 (due to the presence of \(\frac{x}{2}\)). Moreover, the output values will still lie in \(-\frac{\pi}{2}\leq y \leq \frac{\pi}{2}\). However, this function will increase at a slower rate compared to \(y = \arcsin x\) because it is divided by 2.

Step 3: Graph the Function using a Graphing Utility

Now, plot the values for this function by inputting the function into the graphing utility. Ensure the x-values are varying from -2 to 2 and note how y-values will be from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). The shape should represent an increasing function which starts at point (-2,-\(\frac{\pi}{2}\)) and ends at point (2,\(\frac{\pi}{2}\)).

Key Concepts

Arcsin Function

The arcsine function, denoted as \( y = \text{arcsin}(x) \), is a fundamental concept in trigonometry. It's the inverse of the sine function, which means that it tells us the angle whose sine is a given number. For example, if \( \text{sin}(\theta) = x \), then \( \text{arcsin}(x) = \theta \). Understanding this relationship is critical as it allows us to solve equations where the angle is unknown.

One of the unique aspects of the arcsin function is that it only takes values between -1 and 1, since these are the limits within which the sine function operates. Visually, the graph of \( y = \text{arcsin}(x) \) looks like an 'S' laying on its side, with a flat slope at the center that curves up and down towards its ends. It is important to remember that due to the nature of the sine function, the arcsin function is not defined outside the interval [-1, 1].

Furthermore, the function is increasing within its domain; this tells us that as x increases, the value of \( \text{arcsin}(x) \) also increases. This increasing nature reflects the principle that as the sine of an angle increases, so does the angle itself within the defined range.

Domain and Range of Arcsine

The domain of a function refers to the set of all possible input values (x-values), while the range refers to the possible output values (y-values). In the case of the arcsin function, the domain is all real numbers between -1 and 1, inclusive. This restriction is because sine values cannot exceed this range, and since arcsine is the inverse, it inherits this limitation.

The range of \( y = \text{arcsin}(x) \) is between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), inclusive. Why is this range significant? These values correspond to the interval of angles in radians that a sine function can produce, which lies between the bottom and top of the unit circle. So, this range guarantees that any arcsine value is an angle that can be plotted on the unit circle.

When dealing with functions such as \( y = \text{arcsin}(\frac{x}{2}) \), the same principles apply to the domain and range. The revised domain becomes -2 ≤ x ≤ 2 because of the modified input. Nevertheless, the range remains unchanged since the value inside the arcsin function is still bound by -1 and 1 after the division by 2. Understanding domain and range is crucial for graphing and interpreting the behavior of trigonometric functions.

Graphing Utility Usage in Trigonometry

Graphing utilities have become an indispensable tool in studying trigonometry, as they allow for a precise and quick representation of complex functions such as the inverse trigonometric functions.

When graphing the arcsin function, one must consider its characteristics and limitations—mainly its domain and range. By setting the correct intervals on the graphing utility, students can observe the function's behavior within its defined constraints. For example, \( y = \text{arcsin}(\frac{x}{2}) \), plotted correctly, will reveal a gentler slope compared to the standard \( y = \text{arcsin}(x) \) because the input to the function is halved, causing the angle change to become more gradual.

Using a graphing utility effectively requires knowledge of these details, as inputting the function without adjusting the window settings might not display the full graph accurately. Subsequent practice with various functions not only solidifies understanding of their individual characteristics but also enhances one’s ability to manipulate and utilize graphing utilities for more advanced analysis within trigonometry and beyond.