Proving a Sinh(x) expression to be an integer

In summary, there are multiple ways to prove that a sinh(x) expression is an integer. These include using the definition of an integer, trigonometric identities, and mathematical induction. However, there are specific values of x that can be used to prove the expression is an integer, and it is not possible for it to be an integer for all values of x.
  • #1
supersticky99
1
0
Prove or disprove: sinh2(ln([tex]\sqrt{}2[/tex]+[tex]\sqrt{}3[/tex])) is an integerObviously, I used my calc to figure out that the answer is 2. Proving it w/o a calc is hard though.

The Attempt at a Solution



I've tried rewriting sinh2 as (1/4)(e2x+e-2x-2) and after all the substitutions and log rules I get (1/4)(ln([tex]\sqrt{}2[/tex]+[tex]\sqrt{}3[/tex])+1/(ln([tex]\sqrt{}2[/tex]+[tex]\sqrt{}3[/tex]))-2)

what now?...
 
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  • #2
Use the identity

[tex]
\exp{(\ln{x})}=x
[/tex]

to eliminate exp and ln.

ehild
 

Related to Proving a Sinh(x) expression to be an integer

1. How do you prove that a sinh(x) expression is an integer?

The simplest way to prove that a sinh(x) expression is an integer is to use the definition of an integer, which is any whole number without a decimal or fractional component. Plug in different values for x and see if the resulting expression is a whole number. If it is, then the sinh(x) expression is an integer.

2. Can you use trigonometric identities to prove a sinh(x) expression is an integer?

Yes, you can use trigonometric identities to prove a sinh(x) expression is an integer. For example, you can use the identity sinh(x) = (e^x - e^-x)/2 and the fact that e^x and e^-x are both always integers to show that the resulting expression is also an integer.

3. What other methods can be used to prove a sinh(x) expression is an integer?

Another method to prove a sinh(x) expression is an integer is by using mathematical induction. This involves proving that the expression is an integer for a base case, and then showing that if it holds for a certain value of x, it also holds for the next consecutive value of x. If this can be shown to be true for all possible values of x, then the expression is an integer.

4. Are there any specific values of x that can be used to prove a sinh(x) expression is an integer?

Yes, there are specific values of x that can be used to prove a sinh(x) expression is an integer. For example, if x = 0, the expression becomes sinh(0) = (e^0 - e^0)/2 = 0, which is an integer. Similarly, for x = π, the expression becomes sinh(π) = (e^π - e^-π)/2 = (23.1407 - 0.0432)/2 = 11.0488, which is also an integer.

5. Can a sinh(x) expression be an integer for all values of x?

No, a sinh(x) expression cannot be an integer for all values of x. For example, if x = ln(2), the expression becomes sinh(ln(2)) = (e^ln(2) - e^-ln(2))/2 = (2 - 1/2)/2 = 3/4, which is not an integer. Therefore, it is not possible for a sinh(x) expression to be an integer for all values of x.

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