Problem 4 - Algebra 2 and Trigonometry

Question

What is the maximum value of \(y\) on the graph of \(y=\sin x ?\)

Step-by-Step Solution

Answer

The maximum value of \(y\) on the graph of \(y = \sin x\) is 1.

Step 1: Understand the Sine Function

The function \(y = \sin x\) represents a sine wave, which is a periodic function with a wave-like pattern. The range of the sine function is traditionally between -1 and 1.

Step 2: Identify Key Features of the Sine Function

The sine function reaches its maximum value whenever the angle \(x\) is an odd multiple of \(\pi/2\), i.e., at \(x = \frac{\pi}{2} + 2n\pi\) for \(n\) as an integer.

Step 3: Determine the Maximum Value

Since the range of \(y = \sin x\) is from -1 to 1, and we know the sine function does not exceed these values, the maximum value \(y\) can achieve is 1.

Key Concepts

Sine Function

The sine function, represented as \(y = \sin x\), is a fundamental mathematical concept that arises in trigonometry. This function results in a smooth wave-like graph, known as a sine wave. Understanding the sine function helps in many areas such as physics, engineering, and even music.
\(\sin x\) describes how the ratio of the opposite side to the hypotenuse in a right triangle changes as the angle \(x\) changes. The graph of the sine function is continuous and oscillates like a wave.

• Range: The sine function has a range from -1 to 1. This means that whatever angle \(x\) you plug into \(\sin x\), the resulting value for \(y\) will always be between -1 and 1.
• Wavelength: A full period of the sine wave (from 0 to \(2\pi\)) represents one complete cycle.

Learning the properties of the sine function is essential for understanding the periodic nature of trigonometric functions and their applications.

Maximum Value

For any function, determining the maximum value gives insight into the behavior of the function. With the sine function, this involves finding when \(\sin x\) reaches its peak value of 1.
When we analyze the sine function, it attains its maximum of 1 at specific points. This occurs at angles \(x = \frac{\pi}{2} + 2n\pi\), where \(n\) is an integer. These points correspond to when the wave is at its highest point above the horizontal axis.

• The value of 1 represents the peak height of each wave crest in the sine function.
• Since \(\sin x\) is symmetric about the horizontal axis, its minimum value is -1, reached halfway between the maximum points.

Recognizing where a function achieves its maximum value is a critical skill in mathematical analysis and applications, particularly for trigonometric functions like sine.

Periodic Functions

Periodic functions, like the sine function, repeat their values in regular intervals. This periodicity is one of the essential attributes of trigonometric functions and allows these functions to be used for modeling cyclical phenomena such as sound waves, tides, and seasonal patterns.
The sine function is periodic with a period of \(2\pi\). This means that every \(2\pi\) units along the x-axis, the sine function repeats its pattern.

• Period: The distance required to complete one full cycle of the function. For \(\sin x\), the period is \(2\pi\).
• Repetition: Due to the periodic nature, patterns encountered in one cycle will occur in each subsequent cycle, providing predictability in the function's behavior.

Understanding periodic functions is crucial for analyzing systems that exhibit regular intervals or phases, making them indispensable tools in various scientific and engineering applications.