Use the Trigonometric Identity

Oct 13, 2025 by windailys.com 77 views

Introduction

In this article, we will explore the problem of finding the maximum value of the expression $\sin a + \sin b + \sin c + \sin d + \sin e$ subject to the constraint that $a + b + c + d + e = 2\pi$ . This problem is a classic example of an optimization problem, where we need to find the maximum or minimum value of a function subject to certain constraints.

Background and Motivation

The problem of finding the maximum value of a trigonometric expression subject to a constraint is a well-known problem in mathematics. It has been studied extensively in the field of optimization and has many applications in various fields such as physics, engineering, and economics.

Problem Statement

Let $a, b, c, d, e$ be real numbers such that $a + b + c + d + e = 2\pi$ . Find the maximum of $\sin a + \sin b + \sin c + \sin d + \sin e$ .

Solution

To solve this problem, we can use the following approach:

Use the Trigonometric Identity: We can use the trigonometric identity $\sin x = \cos (\frac{\pi}{2} - x)$ to rewrite the expression $\sin a + \sin b + \sin c + \sin d + \sin e$ in terms of cosine functions.
Apply the Cauchy-Schwarz Inequality: We can apply the Cauchy-Schwarz inequality to the expression $\cos a + \cos b + \cos c + \cos d + \cos e$ to obtain a lower bound on the expression.
Use the Equality Case: We can use the equality case of the Cauchy-Schwarz inequality to find the values of $a, b, c, d, e$ that maximize the expression.

Step 1: Use the Trigonometric Identity

We can use the trigonometric identity $\sin x = \cos (\frac{\pi}{2} - x)$ to rewrite the expression $\sin a + \sin b + \sin c + \sin d + \sin e$ in terms of cosine functions:

\sin a + \sin b + \sin c + \sin d + \sin e = \cos (\frac{\pi}{2} - a) + \cos (\frac{\pi}{2} - b) + \cos (\frac{\pi}{2} - c) + \cos (\frac{\pi}{2} - d) + \cos (\frac{\pi}{2} - e)

Step 2: Apply the Cauchy-Schwarz Inequality

We can apply the Cauchy-Schwarz inequality to the expression $\cos a + \cos b + \cos c + \cos d + \cos e$ to obtain a lower bound on the expression:

\left( \cos a + \cos b + \cos c + \cos d + \cos e \right)^2 \leq 5 \left( \cos^2 a + \cos^2 b + \cos^2 c + \cos^2 d + \cos^2 e \right)

Step 3: Use the Equality Case

We can use the equality case of the Cauchy-Schwarz inequality to find the values of $a, b, c, d, e$ that maximize the expression:

\cos a + \cos b + \cos c + \cos d + \cos e = \sqrt{5} \left( \cos^2 a + \cos^2 b + \cos^2 c + \cos^2 d + \cos^2 e \right)^{\frac{1}{2}}

Conclusion

In this article, we have shown that the maximum value of the expression $\sin a + \sin b + \sin c + \sin d + \sin e$ subject to the constraint that $a + b + c + d + e = 2\pi$ is $\sqrt{5}$ . We have used the trigonometric identity, the Cauchy-Schwarz inequality, and the equality case to derive this result.

Alternative Proof

There is an alternative proof of this result that uses the following approach:

Use the Trigonometric Identity: We can use the trigonometric identity $\sin x = \cos (\frac{\pi}{2} - x)$ to rewrite the expression $\sin a + \sin b + \sin c + \sin d + \sin e$ in terms of cosine functions.
Apply the Jensen's Inequality: We can apply Jensen's inequality to the expression $\cos a + \cos b + \cos c + \cos d + \cos e$ to obtain a lower bound on the expression.
Use the Equality Case: We can use the equality case of Jensen's inequality to find the values of $a, b, c, d, e$ that maximize the expression.

Step 1: Use the Trigonometric Identity

We can use the trigonometric identity $\sin x = \cos (\frac{\pi}{2} - x)$ to rewrite the expression $\sin a + \sin b + \sin c + \sin d + \sin e$ in terms of cosine functions:

\sin a + \sin b + \sin c + \sin d + \sin e = \cos (\frac{\pi}{2} - a) + \cos (\frac{\pi}{2} - b) + \cos (\frac{\pi}{2} - c) + \cos (\frac{\pi}{2} - d) + \cos (\frac{\pi}{2} - e)

Step 2: Apply Jensen's Inequality

We can apply Jensen's inequality to the expression $\cos a + \cos b + \cos c + \cos d + \cos e$ to obtain a lower bound on the expression:

\cos a + \cos b + \cos c + \cos d + \cos e \leq 5 \cos \left( \frac{a + b + c + d + e}{5} \right)

Step 3: Use the Equality Case

We can use the equality case of Jensen's inequality to find the values of $a, b, c, d, e$ that maximize the expression:

\cos a + \cos b + \cos c + \cos d + \cos e = 5 \cos \left( \frac{a + b + c + d + e}{5} \right)

Conclusion

In this article, we have shown that the maximum value of the expression $\sin a + \sin b + \sin c + \sin d + \sin e$ subject to the constraint that $a + b + c + d + e = 2\pi$ is $\sqrt{5}$ . We have used the trigonometric identity, Jensen's inequality, and the equality case to derive this result.

References

[1] Cauchy-Schwarz Inequality: This inequality is a fundamental result in mathematics that has many applications in various fields.
[2] Jensen's Inequality: This inequality is a fundamental result in mathematics that has many applications in various fields.
[3] Trigonometric Identity: This identity is a fundamental result in mathematics that has many applications in various fields.

Future Work

In this article, we have shown that the maximum value of the expression $\sin a + \sin b + \sin c + \sin d + \sin e$ subject to the constraint that $a + b + c + d + e = 2\pi$ is $\sqrt{5}$ . In the future, we can explore other optimization problems that involve trigonometric expressions and constraints.

Conclusion

Q: What is the maximum value of the expression $\sin a + \sin b + \sin c + \sin d + \sin e$ subject to the constraint that $a + b + c + d + e = 2\pi$ ?

A: The maximum value of the expression $\sin a + \sin b + \sin c + \sin d + \sin e$ subject to the constraint that $a + b + c + d + e = 2\pi$ is $\sqrt{5}$ .

Q: How did you derive this result?

A: We used the trigonometric identity $\sin x = \cos (\frac{\pi}{2} - x)$ to rewrite the expression $\sin a + \sin b + \sin c + \sin d + \sin e$ in terms of cosine functions. We then applied the Cauchy-Schwarz inequality to the expression $\cos a + \cos b + \cos c + \cos d + \cos e$ to obtain a lower bound on the expression. Finally, we used the equality case of the Cauchy-Schwarz inequality to find the values of $a, b, c, d, e$ that maximize the expression.

Q: What is the alternative proof of this result?

A: The alternative proof of this result uses the trigonometric identity $\sin x = \cos (\frac{\pi}{2} - x)$ to rewrite the expression $\sin a + \sin b + \sin c + \sin d + \sin e$ in terms of cosine functions. We then applied Jensen's inequality to the expression $\cos a + \cos b + \cos c + \cos d + \cos e$ to obtain a lower bound on the expression. Finally, we used the equality case of Jensen's inequality to find the values of $a, b, c, d, e$ that maximize the expression.

Q: What are the applications of this result?

A: This result has many applications in various fields such as physics, engineering, and economics. For example, it can be used to find the maximum value of a trigonometric expression subject to a constraint, which is a common problem in optimization.

Q: What are the limitations of this result?

A: This result assumes that the values of $a, b, c, d, e$ are real numbers. If the values of $a, b, c, d, e$ are complex numbers, then the result may not hold.

Q: Can you provide more examples of optimization problems that involve trigonometric expressions and constraints?

A: Yes, here are a few examples:

Maximum of $\sin a + \sin b + \sin c$ subject to $a + b + c = \frac{\pi}{2}$ : This problem involves finding the maximum value of the expression $\sin a + \sin b + \sin c$ subject to the constraint that $a + b + c = \frac{\pi}{2}$ .
Maximum of $\cos a + \cos b + \cos c$ subject to $a + b + c = \frac{\pi}{2}$ : This problem involves finding the maximum value of the expression $\cos a + \cos b + \cos c$ subject to the constraint that $a + b + c = \frac{\pi}{2}$ .
Maximum of $\sin a + \sin b + \sin c + \sin d$ subject to $a + b + c + d = \pi$ : This problem involves finding the maximum value of the expression $\sin a + \sin b + \sin c + \sin d$ subject to the constraint that $a + b + c + d = \pi$ .

Q: Can you provide more information about the Cauchy-Schwarz inequality and Jensen's inequality?

A: Yes, here are a few more details about the Cauchy-Schwarz inequality and Jensen's inequality:

Cauchy-Schwarz Inequality: This inequality states that for any vectors $\mathbf{x}$ and $\mathbf{y}$ in an inner product space, we have $\left( \mathbf{x} \cdot \mathbf{y} \right)^2 \leq \left( \mathbf{x} \cdot \mathbf{x} \right) \left( \mathbf{y} \cdot \mathbf{y} \right)$ .
Jensen's Inequality: This inequality states that for any convex function $f(x)$ and any random variable $X$ with expected value $E[X]$ , we have $f(E[X]) \leq E[f(X)]$ .

Q: Can you provide more information about the trigonometric identity $\sin x = \cos (\frac{\pi}{2} - x)$ ?

A: Yes, here are a few more details about the trigonometric identity $\sin x = \cos (\frac{\pi}{2} - x)$ :

Trigonometric Identity: This identity states that for any angle $x$ , we have $\sin x = \cos (\frac{\pi}{2} - x)$ .
Proof: The proof of this identity involves using the definition of the sine and cosine functions and the properties of the trigonometric functions.

Q: Can you provide more information about the optimization problems that involve trigonometric expressions and constraints?

A: Yes, here are a few more details about the optimization problems that involve trigonometric expressions and constraints:

Optimization Problems: These problems involve finding the maximum or minimum value of a function subject to certain constraints.
Trigonometric Expressions: These expressions involve the sine and cosine functions and are commonly used in optimization problems.
Constraints: These constraints involve the values of the variables in the optimization problem and are used to restrict the possible solutions.

Conclusion

In this article, we have provided a Q&A section on the maximum value of the expression $\sin a + \sin b + \sin c + \sin d + \sin e$ subject to the constraint that $a + b + c + d + e = 2\pi$ . We have also provided information about the Cauchy-Schwarz inequality, Jensen's inequality, the trigonometric identity $\sin x = \cos (\frac{\pi}{2} - x)$ , and the optimization problems that involve trigonometric expressions and constraints.