Understanding the Maximum Value of sin(x)cos(x): A Comprehensive Guide

The function sin(x)cos(x) is a common topic in trigonometry, often explored in calculus and algebra, especially when discussing the behavior of trigonometric functions in various applications. One fundamental aspect is determining the maximum value of this function. In this guide, we delve into the mathematics behind this function and provide a detailed method to find its utmost peak while aligning with Google's SEO standards.

The Trigonometric Identity and Maximum Value

To begin, we can use the trigonometric identity to simplify the expression sin(x)cos(x). The identity in question is:

sin(x)cos(x) frac{1}{2} sin(2x)

This identity is derived from the product-to-sum formulas and simplifies the process of finding the maximum value. The function sin(2x) oscillates between -1 and 1, meaning the maximum value of sin(2x) is 1. Therefore, the function sin(x)cos(x) will reach its maximum when 2x frac{π}{2} 2kπ, where k is any integer. Simplifying this, we find that the maximum value of sin(x)cos(x) is:

frac{1}{2} * 1 frac{1}{2}

Deriving the Maximum Value through Derivatives

Alternatively, we can use calculus to find the maximum value. By taking the derivative of the function y sin(x)cos(x) and setting it to zero, we can locate the critical points and determine the maximum value.

y sin(x)cos(x)
dy/dx cos(2x)

To find the maximum, we solve the equation cos(2x) 0. This occurs when 2x frac{2n-1}{2}π, where n is an integer. Geometrically, this corresponds to the points where the slope of sin(2x) changes from positive to negative. Plotting this on a graph, we see that the function reaches its maximum at these points, which are the extreme maxima and minima.

It is also worth noting that the combined effect of sin(x) and cos(x) is such that their product is maximized when both functions are at their maximum relative values simultaneously. This happens at the angle of x 45° (or frac{π}{4} radians), where both sin(x) and cos(x) are equal to frac{sqrt{2}}{2}. Therefore, the maximum value of sin(45°)cos(45°) is:

frac{sqrt{2}}{2} * frac{sqrt{2}}{2} frac{1}{2}

Geometric Insight

Another perspective is to consider the geometric interpretation of sin(x)cos(x). Since both sin(x) and cos(x) oscillate between -1 and 1, the product sin(x)cos(x) will be maximized when both functions are at their maximum values simultaneously. The maximum of sin(x) is 1, and the maximum of cos(x) is also 1. Thus, the maximum value of sin(x)cos(x) is the product of these maxima divided by 2:

frac{1}{2}

Conclusion

In conclusion, the maximum value of sin(x)cos(x) is frac{1}{2}. This value is achieved when either x 45° (or frac{π}{4}) or when the function reaches its critical points via calculus. Understanding the trigonometric identity and applying calculus methods can help us derive this important value.

By following these steps and aligning our content with Google's SEO standards, we can ensure that this guide is easily discoverable and provides a well-rounded explanation of the maximum value of sin(x)cos(x).