Introduction to the Maximum Value of 6sin(x)cos(x)

Understanding the maximum value of trigonometric functions is crucial in many fields, including physics, engineering, and data analysis. This article explores the specific example of the function 6sin(x)cos(x) to demonstrate the application of trigonometric identities and calculus concepts. This knowledge is instrumental for SEO optimization, ensuring that your content aligns with Google's standards for quality and relevance.

Understanding the Function

The function we are analyzing is fx 6sin(x)cos(x). On the surface, this function might seem complex, but it can be simplified using a well-known trigonometric identity:

Using the Double Angle Identity

The double angle identity for sine states that sin(2x) 2sin(x)cos(x). Applying this identity to our function:

6sin(x)cos(x) 3sin(2x)

This simplification makes it much easier to analyze the function. The sine function sin(2x) has a maximum value of 1, which allows us to determine the maximum value of 6sin(x)cos(x) straightforwardly.

Determining the Maximum Value

The maximum value of 3sin(2x) is 3, as the sine function ranges between -1 and 1. Therefore, the maximum value of the original function 6sin(x)cos(x) is:

fxmax 3

Deriving the Maximum Using Calculus

For a more detailed analysis, we can use calculus to find the critical points where the function's derivative equals zero. The derivative of 6sin(x)cos(x) is:

f'(x) 6cos(2x)

Solving 6cos(2x) 0 gives us:

cos(2x) 0 2x dfrac{pi}{2} npi, n in mathbb{Z} x dfrac{pi}{4} dfrac{npi}{2}, n in mathbb{Z}

Testing the intervals around these points, we find that the function has a maximum value when:

x dfrac{pi}{4} dfrac{npi}{2}, n in mathbb{Z}

For n 0, the function reaches its maximum value of 3 at x dfrac{pi}{4}.

Conclusion and Optimization Tips

In summary, the function 6sin(x)cos(x) attains its maximum value of 3 when analyzed using trigonometric identities and calculus techniques. This example highlights the importance of combining different mathematical approaches to solve complex problems. For SEO optimization, ensure your content is clear, concise, and relevant to both users and search engines. Use headers, subheadings, and logical structure to make your article more readable and easier to comprehend.

Key Takeaways:

The double angle identity simplifies the analysis of trigonometric functions. Derivatives can be used to find critical points where the function reaches its maximum or minimum. Using mathematical principles effectively enhances the relevance of content for search engines like Google.

By incorporating these strategies, you can improve the SEO performance of your content and engage your audience effectively.