Maximizing the Value of sin(x)cos(x): Techniques and Proofs

The function sin(x)cos(x) is a fundamental trigonometric expression that often appears in various mathematical problems and physical scenarios. Understanding the maximum value of this expression is crucial for many applications in mathematics, physics, and engineering. In this article, we will explore different methods to find the maximum value of sin(x)cos(x) and provide detailed proofs.

Trigonometric Identity and Calculus Approach

To find the maximum value of sin(x)cos(x), we can use a trigonometric identity. Specifically, we can rewrite the expression as follows:

(sin x cos x sqrt{2} left( frac{1}{sqrt{2}} sin x frac{1}{sqrt{2}} cos x right) sqrt{2} sin left( x frac{pi}{4} right))

Finding the Maximum

The function (sin(x)) has a maximum value of 1. Therefore, the maximum value of (sqrt{2} sin left( x frac{pi}{4} right)) is (sqrt{2} cdot 1 sqrt{2}). This maximum value occurs when (x frac{pi}{4} frac{pi}{2} 2kpi) for any integer (k), or equivalently, when (x frac{pi}{4} 2kpi).

Alternative Proof Using Inequality

To further solidify our understanding, we can use the inequality (2ab le a^2 b^2) to prove the maximum value. Starting from the expression (sin^2 x cos^2 x 1), we can manipulate it as follows:

[sin x cos x le frac{1}{2} (sin^2 x cos^2 x) frac{1}{2}]

Multiplying both sides by 2, we get:

[2sin x cos x le 1]

Therefore:

[sin x cos x le frac{1}{2} cdot 2 sqrt{2}]

Hence, the maximum value of (sin(x)cos(x)) is (sqrt{2}).

Non-Calculus Method Using Angle Addition Rules

My favorite approach to solve this problem without using calculus is through the angle addition formula:

(sin(A - B) sin A cos B - cos A sin B)

Setting (B frac{pi}{4}) and knowing that (cos frac{pi}{4} sin frac{pi}{4} frac{1}{sqrt{2}}), we can rewrite:

(sin x cos x sqrt{2} sin (x - frac{pi}{4}))

This expression clearly has a maximum value of (sqrt{2}) when (sin (x - frac{pi}{4}) 1), which occurs when (x - frac{pi}{4} frac{pi}{2} 2npi) for some integer (n).

Thus, the greatest value that (sin x cos x) can have is (sqrt{2}) or approximately 1.414.