Range of the Function f(x) sin^6(x) cos^6(x) 1: A Detailed Analysis
The function f(x) sin^6(x) cos^6(x) 1 presents an interesting mathematical challenge, particularly in determining its range. This article delves into the detailed derivation of the range by transforming the function into a more manageable form, and provides a thorough explanation through logical steps and identities.
Key Identities and Transformations
In our analysis, we will utilize fundamental trigonometric identities to simplify and transform the function f(x) sin^6(x) cos^6(x) 1. The initial steps involve expressing the sixth powers of sine and cosine in terms of their squares.
Given:
[f(x) sin^6(x) cos^6(x) 1]First, we express sin^6(x) cos^6(x) as:
[sin^6(x) cos^6(x) (sin^2(x) cos^2(x))^3]Using the identity for the cube of a difference, a^3 - b^3 (a - b)(a^2 ab b^2), and setting a sin^2(x) and b cos^2(x), we get:
[(sin^2(x) cos^2(x))^3 (sin^2(x) - cos^2(x))(sin^2(x) cos^2(x))^2 (sin^2(x) cos^2(x))]Multiplying out the terms, we have:
[sin^6(x) cos^6(x) sin^2(x) cos^2(x) (sin^2(x) cos^2(x) - 2 sin^2(x) cos^2(x) cos^2(x) sin^2(x))]Simplifying further, we get:
[sin^6(x) cos^6(x) sin^2(x) cos^2(x) (sin^2(x) cos^2(x) - 2 sin^2(x) cos^2(x) cos^2(x) sin^2(x)) sin^2(x) cos^2(x) (1 - 3 sin^2(x) cos^2(x))]Expressing sin^2(x) cos^2(x) in terms of a single sine function, we use the identity sin^2(x) cos^2(x) frac{1}{4} sin^2(2x), thus:
[sin^6(x) cos^6(x) frac{1}{4} sin^2(2x) (1 - 3 cdot frac{1}{4} sin^2(2x))]Simplifying, we get:
[sin^6(x) cos^6(x) frac{1}{4} sin^2(2x) (1 - frac{3}{4} sin^2(2x))]Therefore, the function f(x) sin^6(x) cos^6(x) 1 can be written as:
[f(x) 1 - frac{3}{4} sin^2(2x) 1 2 - frac{3}{4} sin^2(2x)]Determining the Range
To determine the range, we need to consider the range of - frac{3}{4} sin^2(2x). As 0 le sin^2(2x) le 1, the term sin^2(2x) varies between 0 and 1. Thus, the term frac{3}{4} sin^2(2x) varies between 0 and frac{3}{4}. Consequently, the term - frac{3}{4} sin^2(2x) varies between 0 and - frac{3}{4}.
Substituting back, we have:
[2 - frac{3}{4} sin^2(2x) in left[2 - frac{3}{4}, 2right] left[frac{5}{4}, 2right]]Conclusion
The range of the function f(x) sin^6(x) cos^6(x) 1 is:
[left[frac{5}{4}, 2right]]This derivation demonstrates the use of trigonometric identities and logical steps to simplify complex functions, leading to a clear understanding of their ranges. The function f(x) sin^6(x) cos^6(x) 1 has a range from frac{5}{4} to 2, which can be valuable in various mathematical and applied contexts.