Proving Inequalities Involving Sine Functions

When dealing with mathematical expressions involving sine functions, such as sin(x^3), sin(y^3), sin(z^3), sin(-xyz), proofs can be quite intricate. In this article, we explore the bounds of these expressions and how to prove that under certain conditions, they cannot reach a specific value.

Introduction

The sine function, denoted as sin(θ), takes values ranging from -1 to 1 for any real number . This property forms the foundation for understanding how sums and differences of sine functions behave. Often, we need to prove inequalities involving these functions, which can be challenging. In this article, we will discuss a proof for a specific case where we need to show that a sum of sine functions cannot reach a certain value.

Sine Functions and Their Ranges

First, let's revisit the range of the sine function:

Range of Sine Function

The range of the sine function is [-1, 1]. This means that for any sine term in an expression, its value will always lie between -1 and 1. When we consider a sum of multiple sine terms, the overall sum is also bounded by these limits.

Proof: Sum of Sine Functions

The given expressions are:

sin(x^3) sin(y^3) sin(z^3) sin(-xyz)

We aim to prove that the sum of these terms, under certain conditions, cannot equal 4. Let's break down the proof step-by-step:

Step 1: Expressions and Their Ranges

For any θ, the sine function has the property:

-sin(θ) sin(-θ)

Using this property, we can rewrite sin(-xyz) as sin(xyz). Therefore, the expression now becomes:

sin(x^3) sin(y^3) sin(z^3) sin(xyz)

Each of these sine functions has a range of [-1, 1]. The sum of these terms can therefore range from -4 to 4.

Step 2: Finding the Maximum Sum

To prove that the sum of these sine functions cannot equal 4, we need to find the scenarios where the sum is maximized and show that these scenarios do not reach 4.

Consider the case where:

sin(x^3) 1 sin(y^3) 1 sin(z^3) 1

For sin(θ) to be 1, we have:

x^3 y^3 z^3 π/2 2kπ

Here, k is an integer. This implies:

x (π/2 2kπ)^(1/3) y (π/2 2kπ)^(1/3) z (π/2 2kπ)^(1/3)

With these conditions, the sum of the first three sine terms is:

1 1 1 3

Step 3: Analyzing the Last Term

The last term, sin(xyz), needs to be analyzed further. Since x, y, z (π/2 2kπ)^(1/3), we have:

xyz (π/2 2kπ)

For this expression to match the form of a sine function, we need:

xyz (π/2 2kπ) π/2 mod 2π

Therefore:

sin(xyz) sin(π/2) 1

Step 4: Contradiction

Now, the overall sum of the sine terms is:

3 1 4

However, let's analyze the condition -sin(xyz) 1 more deeply:

-sin(xyz) 1 implies:

sin(xyz) -1

For this to be true, we need:

xyz -π/2 2kπ

Substituting xyz (π/2 2kπ), we get:

(π/2 2kπ) -π/2 2kπ

which simplifies to:

π/2 -π/2

This contradiction shows that it is impossible to satisfy the condition for -sin(xyz) 1.

Conclusion

Therefore, the sum of the sine functions sin(x^3), sin(y^3), sin(z^3), sin(-xyz) cannot reach the value of 4. The range of the sum is strictly between -4 and 4, excluding 4 itself.

Key Takeaways

The range of the sine function is [-1, 1]. The sum of sine terms is bounded by these limits. Proving the sum cannot reach 4 involves analyzing the conditions under which each term can take its maximum or minimum value.

Further Reading

If you are interested in more in-depth discussions and proofs involving trigonometric functions and inequalities, you may want to explore the works of seasoned mathematicians like Alon Amit and David Joyce.