Maximizing the Product of Sine and Cosine to the Fourth Power

Understanding and finding the maximum value of sin^4(x) cos^4(x) involves a combination of trigonometric identities and optimization techniques. This article delves into the process and the key concepts involved.

Understanding the Problem

Given the function sin^4(x) cos^4(x), we aim to determine the maximum value of this expression over all possible values of x.

Trigonometric Identity Application

To begin, we can leverage a well-known trigonometric identity. Recall that:

sin^4(x) cos^4(x) (sin^2(x) cos^2(x))^2

This can be further broken down using another identity:

sin^2(x) cos^2(x) 1 - (sin^2(x) - cos^2(x))^2

But, for simplicity, we can use:

sin^2(x) cos^2(x) 0.5 (sin(2x))^2

Thus, we can write:

sin^4(x) cos^4(x) 0.25 (sin(2x))^4

Optimization via Completing the Square

Next, we consider the function 0.25 (sin(2x))^4 and attempt to find its maximum value. The function (sin(2x))^4 is always non-negative because the sine function returns values between -1 and 1, and squaring and then cubing these values yields a non-negative result.

To analyze (sin(2x))^4, let's use the parameter y sin(2x). Therefore:

0.25 (sin(2x))^4 0.25 y^4

We need to find the maximum value of 0.25 y^4 for y in the interval [-1, 1].

Notice that y^4 achieves its maximum value when y±1. Therefore:

0.25 (1)^4 0.25 (1) 0.25

Evaluating the Expression at Key Points

Let's evaluate sin^4(x) cos^4(x) at specific points:

sin^4(0) cos^4(0) (0)^4 (1)^4 0

sin^4(π/2) cos^4(π/2) (1)^4 (0)^4 0

sin^4(π/4) cos^4(π/4) (1/√2)^4 (1/√2)^4 (1/4)^2 1/16

We can generalize by recognizing the symmetry and periodicity of the sine and cosine functions. The maximum value of sin(2x) is 1, which occurs at x π/4 nπ/2 for integer values of n. Therefore, the maximum value of sin^4(x) cos^4(x) is:

maximum value of sin^4(x) cos^4(x) 0.25

Conclusion

The maximum value of the product sin^4(x) cos^4(x) is 0.25, which occurs at key points such as x π/4 nπ/2. Understanding this process involves a deep dive into trigonometric identities and optimization techniques, providing a rich context for further exploration in mathematical problem-solving.