Finding The Maximum Value Of Sin(x) A Comprehensive Guide

The maximum value of a function is a fundamental concept in mathematics, particularly in calculus and trigonometry. When dealing with trigonometric functions like the sine function, understanding its behavior is crucial for solving various problems. In this comprehensive exploration, we will delve into the function f(x) = sin(x), meticulously analyzing its properties to pinpoint its maximum value. We will navigate through the sine wave, examining its oscillations and key characteristics to definitively answer the question: What is the maximum of f(x) = sin(x)?

Understanding the Sine Function

Before we dive into finding the maximum, let's establish a solid understanding of the sine function itself. The sine function, denoted as sin(x), is a trigonometric function that relates an angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. In the context of a unit circle (a circle with a radius of 1), the sine of an angle x is represented by the y-coordinate of the point where the terminal side of the angle intersects the circle. This geometric interpretation is crucial for visualizing the sine function's behavior.

The Sine Wave

When we graph the sine function, we obtain a wave-like curve that oscillates between -1 and 1. This sine wave is a visual representation of how the sine value changes as the angle x varies. The x-axis represents the angle (typically in radians), and the y-axis represents the sine value, sin(x). The wave starts at 0 when x is 0, rises to a peak, falls back to 0, reaches a trough, and then returns to 0, completing one full cycle. This cyclical behavior is a defining characteristic of the sine function.

Key Characteristics of the Sine Function

Several key characteristics of the sine function are essential for determining its maximum value:

• Amplitude: The amplitude of the sine function is the distance from the midline (the x-axis in the case of f(x) = sin(x)) to the peak or trough of the wave. For f(x) = sin(x), the amplitude is 1, meaning the wave oscillates 1 unit above and 1 unit below the x-axis.
• Period: The period is the length of one complete cycle of the wave. For f(x) = sin(x), the period is 2π, meaning the wave repeats itself every 2π radians.
• Range: The range of the sine function is the set of all possible output values. For f(x) = sin(x), the range is [-1, 1], indicating that the sine value can be any real number between -1 and 1, inclusive.

Determining the Maximum Value of sin(x)

Now that we have a firm grasp of the sine function's properties, we can pinpoint its maximum value. As we observed earlier, the sine wave oscillates between -1 and 1. This means that the highest point the wave reaches corresponds to the maximum value of the function. From the graph and our understanding of the unit circle, we can see that the sine function reaches its maximum value when the angle x corresponds to the point where the terminal side intersects the unit circle at its highest point. This occurs when the angle is π/2 radians (90 degrees).

The Maximum Value

At x = π/2, the y-coordinate on the unit circle is 1. Therefore, sin(π/2) = 1. This is the maximum value of the sine function. The sine function never exceeds 1, and it attains this value at infinitely many points, including π/2, π/2 + 2π, π/2 + 4π, and so on. These points correspond to angles that are coterminal with π/2.

Why Other Options are Incorrect

Let's address why the other options provided in the original question are incorrect:

• A. -2π: This value is not within the range of the sine function. The sine function's output is always between -1 and 1.
• B. -1: This is the minimum value of the sine function, not the maximum. The sine function reaches -1 at angles such as 3π/2.
• D. 2π: This value is also outside the range of the sine function. While 2π is the period of the sine function, it does not represent its maximum value.

Conclusion

In conclusion, the maximum value of the function f(x) = sin(x) is 1. This value is attained when the angle x is π/2 radians (90 degrees) or any angle coterminal with π/2. Understanding the sine function's properties, such as its amplitude, period, and range, is crucial for accurately determining its maximum value. The sine wave's oscillation between -1 and 1 visually confirms that 1 is the highest point the function reaches, making it the maximum. By grasping these fundamental concepts, we can confidently analyze and solve problems involving trigonometric functions.

This detailed explanation provides a comprehensive understanding of how to find the maximum value of the sine function, not only answering the specific question but also building a strong foundation for further exploration of trigonometric concepts. Remember, the sine function is a cornerstone of mathematics and physics, and mastering its properties is an invaluable skill.