Problem 97 - Precalculus Essentials

Question

In Exercises \(95-98,\) graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, find a value of \(x\) for which both sides are defined but not equal. $$\sin \frac{x}{2}=\frac{1}{2} \sin x$$

Step-by-Step Solution

Answer

The functions \(\sin \frac{x}{2}\) and \(\frac{1}{2} \sin x\) are not identical. An example of a value of \(x\) for which both sides of the equation are defined but unequal is \(x=\pi\).

Step 1: Graphing the Left Side of the Equation

First, graph the function \(\sin \frac{x}{2}\) using a graphing tool. This will produce a wave-like graph, characteristic for the sine function. The graph of \(\sin \frac{x}{2}\) has a period of \(\pi\), and thus appears twice as 'spread out' as the standard sine function.

Step 2: Graphing the Right Side of the Equation

Next, graph the function \(\frac{1}{2} \sin x\). This function is a sine function with half the amplitude of the standard sine function, so it will also produce a wave-like graph, but the wave peaks and troughs will only reach half as high and low as the standard sine function.

Step 3: Comparing the Graphs

Lay the two graphs over each other to compare them. These two functions do not coincide. Therefore, the original equation \(\sin \frac{x}{2}=\frac{1}{2} \sin x\) is not an identity. This is because the two functions have different characteristics: \(\sin \frac{x}{2}\) with a longer period and \(1/2 \sin x\) with a smaller amplitude.

Step 4: Finding a Specific Value of x

Since the graphs of the two functions do not overlap perfectly, it is possible to find a value of \(x\) for which the two sides of the equation yield different results. For example, for \(x= \pi\), the left side yields \( \sin \frac{\pi}{2} = 1\) and the right side yields \( \frac{1}{2} \sin \pi = 0\). Therefore, \(\pi\) is an example of a value for which both sides of the equation are defined but not equal.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, often denoted as \( \sin(x) \). It represents the y-coordinate of a point on a unit circle as the angle \( x \), measured in radians, increases. This function is periodic, which means it repeats its values in regular intervals.

• The standard sine function, \( \sin(x) \), has a period of \( 2\pi \), meaning every \( 2\pi \) units, the function cycle repeats.
• Its amplitude, or the height of the wave peaks from the center line, is 1.
• The range of \( \sin(x) \) spans from -1 to 1, perfectly capturing the oscillatory nature of waves.

Understanding the sine function is crucial because it lays the groundwork for analyzing more complex trigonometric functions. For instance, modifying the input of the sine function, as in \( \sin(\frac{x}{2}) \), alters its behavior by adjusting the period.

Graphing Trigonometric Functions

Graphing trigonometric functions helps us visualize their periodic nature. Let's look at how modifications to the sine function appear on a graph:
When we graph \( \sin \frac{x}{2} \), the function has the same basic sine wave shape, but with adjustments:

• The period is doubled to \( 4\pi \) due to the \( \frac{1}{2} \) factor, stretching the wave.
• The amplitude remains the same as the standard sine function, which is 1.

Comparatively, \( \frac{1}{2} \sin x \) is another sine function variant with unique characteristics:

• The amplitude is halved, meaning the peaks and troughs only reach a maximum and minimum of \( \frac{1}{2} \).
• The period remains \( 2\pi \), identical to the standard sine function.

By graphing these functions in the same viewing rectangle, the visual differences, such as the mismatched period and amplitude, become apparent, illustrating why the equation \( \sin \frac{x}{2} = \frac{1}{2} \sin x \) is not an identity.

Identities in Trigonometry

Trigonometric identities are equations that hold true for all angles within their domains, showcasing fundamental properties of trigonometric functions. Recognizing identities is vital for simplifying expressions and solving equations.
To determine if an equation is an identity, such as testing if \( \sin \frac{x}{2} = \frac{1}{2} \sin x \), we first graph each side. If their graphs overlap completely, the equation is an identity. Otherwise, it is not.
In this case, differences in the graphs reveal the lack of identity caused by:

• \( \sin \frac{x}{2} \) having a longer period.
• \( \frac{1}{2} \sin x \) having a reduced amplitude.

To verify non-identity further, find specific \( x \) values that produce unequal outcomes for each side of the equation. For example, when \( x = \pi \), \( \sin \frac{\pi}{2} = 1 \) whereas \( \frac{1}{2} \sin \pi = 0 \), clearly demonstrating the two sides aren’t equal, solidifying the understanding that the equation is non-identical for all \( x \).