Problem 3 - Algebra 2 and Trigonometry

Question

Find the amplitude of each function. \(y=\sin x\)

Step-by-Step Solution

Answer

The amplitude is 1.

Step 1: Identify the Function Type

First, recognize that the given function is a sine function, noted as \(y = \sin x\). Sine functions are periodic and oscillate between certain values, making them a common trigonometric function to analyze.

Step 2: Understand the Standard Sine Function

In the standard sine function \(y = \sin x\), the amplitude is the maximum distance from the midline of the function to its peak or trough. The standard sine function oscillates between +1 and -1.

Step 3: Determine the Amplitude

The amplitude of a sine function is determined by the coefficient in front of the sine function. In the function \(y = \sin x\), there is an implicit coefficient of 1 in front of \(\sin x\), thus, the amplitude is 1.

Key Concepts

Sine Function

The sine function, denoted as \(y = \sin x\), is one of the fundamental trigonometric functions used to describe a repeating wave pattern. When plotted on a graph, it forms a smooth, wavelike curve, reflecting its periodic nature.

Key characteristics of the sine function include:

• It is cyclical and repeats every \(2\pi\) radians (360 degrees).
• The graph of the sine function undulates between a maximum value of 1 and a minimum value of -1 without stretching or shifting.
• The sine function is continuous, meaning there are no gaps or jumps in its graph.

Understanding the sine function is important as it forms the basis for describing other more complex trigonometric functions. Its behavior is crucial in fields like physics, engineering, and signal processing where wave patterns and harmonic motions are observed.

Amplitude

Amplitude refers to the maximum displacement of the wave from its central position, or midline, in the context of a graph of a trigonometric function like sine.

For the standard sine function, \(y = \sin x\), the amplitude is simply the absolute value of the coefficient in front of the \(\sin x\) term. Here, the function has an implicit coefficient of 1, thus the amplitude is 1.

In mathematical terms, the formula to find the amplitude for a function \(y = a \sin x\) is

• Amplitude = \( |a| \)

This means if the equation was \(y = 3 \sin x\), the amplitude would be 3. The amplitude affects how tall or short the wave appears, giving us insight about the maximum extent the function deviates from its central axis.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or cycles. The sine function is one example of such a function.

Here's what to know about periodic functions:

• The basic characteristic of a periodic function is that there exists a positive number, called the period, such that the function repeats itself after every interval of this length.
• For the sine function, \(y = \sin x\), the period is \(2\pi\) radians. That means every \(2\pi\) units along the x-axis, the sine wave looks exactly the same.
• Periodic functions are vital in mathematical modeling because they provide a simple way to describe cycles or waves, such as tides, sound waves, and even certain market trends.

Understanding how periodic functions work helps us analyze situations in both natural phenomena and engineered systems, making them a cornerstone of mathematical studies.