Understanding the Limit of the Sine Function: Maximum and Minimum Values

The sine function, denoted as sinθ, is a fundamental trigonometric function with a distinctive behavior. It oscillates between -1 and 1, making it an important topic in mathematics and its applications. This article delves into the concept of the maximum and minimum values of the sine function, providing a thorough understanding of its limits.

What is the Maximum Value of the Sine Function?

The maximum value of the sine function occurs at the highest point on its graph. Specifically, the sine function attains its maximum value of 1 at 90 degrees or π/2 radians. This point can be identified by examining the graph or using calculus to find the critical points.

Limits of the Sine Function

To understand the overall behavior of the sine function, it's crucial to recognize its bounded nature. The sine function is defined as

-1 ≤ sinθ ≤ 1

where θ is the angle in radians. This expression indicates that the sine function does not have an upper or lower bound other than these two values. The function oscillates continuously within the interval [-1, 1], making it periodic with a period of 2π.

Graphical Interpretation

By plotting the graph of the sine function, we can visually observe that the maximum value is indeed 1. This graphical representation provides a clear illustration of the sine function's behavior and helps in understanding its properties.

Finding the Maximum and Minimum Values Analytically

To find the maximum and minimum values of a sine function, we can use calculus. The process involves finding the critical points and their nature as maxima or minima using the first and second derivatives of the function.

Step-by-Step Procedure

Step 1: First Derivative Test

Find the first derivative of the function, which in this case is the derivative of sinx. Equate the first derivative to zero to find the critical points. For the sine function, this is given by:

Put the first derivative equal to zero: dxsinx/dx cosx 0

The solutions are

x (2n 1)π/2

Step 2: Second Derivative Test

Calculate the second derivative and evaluate it at the critical points to determine the nature of these points. The second derivative of the sine function is

d2sinx/dx2 -sinx

Evaluating the second derivative at x π/2:

-sin(π/2) -1

Since the second derivative is negative, the point x π/2 represents a maximum. This confirms that the maximum value of the sine function is indeed 1.

Trigonometric Properties

The sine function, sinx, always takes values between -1 and 1. This is a fundamental property of the sine function and is important in various mathematical and scientific applications. For instance,

-1 ≤ sinx ≤ 1

This inequality helps in solving trigonometric equations and simplifying expressions involving the sine function.

Conclusion

In summary, the sine function has a maximum value of 1 and a minimum value of -1. The function is continuous and periodic, oscillating between these values with a period of 2π. By using calculus, specifically the first and second derivative tests, we can find these maximum and minimum values analytically.