Solving the Mystery of sinh(-3) ≥ -3 or sinh(-3) ≤ -3?

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In summary, the conversation discusses using the fact that exp(y) ≥ 1 + y for all y to solve for sinh. It explores an example using y=3 and the inequalities for sinh(3) and sinh(-3). The conversation also points out a mistake made when subtracting the two inequalities and provides a clarification.
  • #1
aaaa202
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I'm given the fact that exp(y) ≥ 1 + y for all y

I will use this for sinh. Since sinh = (exp(y) - exp(-y))/2 we have for y=3 as example:

sinh(3) ≥ 3 by subtracting the two inequalities

Instead for sinh(-3) I get:

sinh(-3) ≥ -3 (1)

But something is wrong here: Because if I take the first inequality sinh(3) ≥ 3 and multiply it by (-1) and switch around the inequality sinh and using the fact that
-sinh(3) = sinh(-3) I get:

sinh(-3) ≤ -3 (2)

This is clearly weird. Which of (1) and (2) is right and why is other one wrong?
 
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  • #2
Careful when subtracting inequalities:

5 > 3
4 > 1

subtracting:
1 > 2 ?
 
  • #3
ahh omg. Thanks LOL
 

FAQ: Solving the Mystery of sinh(-3) ≥ -3 or sinh(-3) ≤ -3?

What is sinh(-3)?

Sinh(-3) is a mathematical function that stands for the hyperbolic sine of -3. It is defined as (e^x - e^-x)/2, where e is Euler's number.

Why is it important to solve the inequality sinh(-3) ≥ -3 or sinh(-3) ≤ -3?

Solving this inequality helps us understand the behavior of the hyperbolic sine function and its relationship to real numbers. It is also a fundamental concept in calculus and other areas of mathematics.

How do you solve the inequality sinh(-3) ≥ -3 or sinh(-3) ≤ -3?

To solve this inequality, we need to use algebraic manipulation and properties of hyperbolic functions. First, we can rewrite the inequality as sinh(-3) - (-3) ≥ 0 or sinh(-3) - (-3) ≤ 0. Then, we can use the fact that sinh(-3) is an odd function, which means sinh(-x) = -sinh(x). This allows us to simplify the inequality to -sinh(3) - 3 ≥ 0 or -sinh(3) - 3 ≤ 0. Finally, we can use the definition of sinh(x) to solve for the values of x that satisfy the inequality.

What are the solutions to the inequality sinh(-3) ≥ -3 or sinh(-3) ≤ -3?

The solutions to this inequality are x ≤ -3 or x ≥ -3. This means that any value of x that is less than or equal to -3 or greater than or equal to -3 will satisfy the inequality. In other words, the solution set is the entire real number line.

How does solving the inequality sinh(-3) ≥ -3 or sinh(-3) ≤ -3 relate to real-life applications?

The hyperbolic sine function is used to model various phenomena in physics, such as the motion of a pendulum or the shape of a hanging chain. Solving this inequality allows us to determine the range of values for which these models are valid. It also has applications in engineering, where hyperbolic functions are used to solve problems involving electric circuits and heat transfer.

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