The Greatest and Least Value of Sinx Cosx in Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the fundamental trigonometric identities involves the product of sinx and cosx. This article will explore the greatest and least values of sinx cosx, providing a comprehensive understanding of its behavior using both graphical and analytical methods.

Introduction

The product of sinx and cosx can be expressed using a double-angle formula, which simplifies the expression significantly:

Mathematical Representation

Using the double-angle identity:
sinxcosx 1/2 sin2x

Graphical Interpretation

Let's consider the equation sinx cosx and plot its graph to visualize the maximum and minimum values. The Double-Angle Identity can be rewritten as:

sinx cosx 1/2 sin2x

Since the maximum and minimum values of sin2x are 1 and -1, respectively, the maximum and minimum values of 1/2 sin2x are:

Maximum value: 1/2 Minimum value: -1/2

Analytical Methods

In addition to graphical methods, we can also use analytical techniques to find the maximum and minimum values of sinx cosx. We will explore two methods:

Method 1: Using the Double-Angle Formula

By expressing sinx cosx as:

sinx cosx 1/2 sin2x

It is evident from the properties of the sine function that the maximum and minimum values of sin2x are 1 and -1, respectively. Therefore, the maximum and minimum values of 1/2 sin2x are:

Maximum value: 1/2 Minimum value: -1/2

Method 2: Using Differentiation

We can also use calculus to find the critical points of sinx cosx. By finding the derivative and setting it to zero:

f(x) sinx cosx

f'(x) cos^2x - sin^2x 0

Solving for x gives us the points where the derivative is zero. These points correspond to the local maxima and minima of the function.

Graph of y sinx cosx 1/2 sin2x

The graph of sinx cosx can be plotted as:

y sinx cosx 1/2 sin2x

From the graph:

-1 sin2x 1 -1/2 ≤ 1/2 sin2x ≤ 1/2

This confirms that the greatest and least values of sinx cosx are 1/2 and -1/2, respectively.

Conclusion

The greatest and least values of the product sinx cosx can be determined using both graphical and analytical methods. These values are crucial for understanding the behavior of trigonometric functions and can be applied in various fields such as physics, engineering, and signal processing.