Optimizing Trigonometric Expressions Using AM-GM Inequality

In this article, we delve into finding the minimum value of the expression 9cos^2x 16sec^2x using the AM-GM inequality. This method is particularly useful for optimizing expressions involving reciprocal trigonometric functions.

Introduction to AM-GM Inequality

The Arithmetic Mean-Geometric Mean (AM-GM) inequality is a fundamental concept in algebra and provides a powerful tool for solving optimization problems. It states that for any set of non-negative real numbers, the arithmetic mean is always greater than or equal to the geometric mean. Mathematically, for any non-negative real numbers (a) and (b), the inequality is expressed as:

frac{a b}{2} geq sqrt{ab}

Applying AM-GM to Trigonometric Expressions

Given the expression 9cos^2x 16sec^2x, we can rewrite sec^2x in terms of cos^2x. Recall that:

sec^2x frac{1}{cos^2x}

Thus, the expression can be transformed as:

y 9cos^2x frac{16}{cos^2x}

where y cos^2x. Here, (y) is restricted to the interval [0, 1] since cos^2x is always non-negative and at most 1.

Determining the Critical Points

To find the minimum value, we take the derivative of the function with respect to (y) and set it to zero.

frac{d}{dy}(9y frac{16}{y}) 9 - frac{16}{y^2}

Setting the derivative to zero:

9 - frac{16}{y^2} 0

Solving for y^2:

y^2 frac{16}{9} Rightarrow y frac{4}{3}

Since (y cos^2x) and must be in the interval [0, 1], the critical point (y frac{4}{3}) is not valid. Therefore, we need to evaluate the function at the endpoints of the interval.

Evaluating the Endpoints

As (y to 0^ ), the value of the function approaches infinity. At the endpoint (y 1):

9(1) frac{16}{(1)^2} 9 16 25

Since the function approaches infinity as (y) approaches 0, and there are no other critical points in the interval, the minimum value of the expression (9cos^2x 16sec^2x) is 25, occurring at (y 1).

Conclusion

Using the AM-GM inequality, we demonstrated that the minimum value of the given trigonometric expression is 25, achieved when cos^2x 1. This solution highlights the power of algebraic inequalities in solving complex optimization problems involving trigonometric functions.

Keywords: AM-GM inequality, trigonometric functions, minimization problem