Question

What is the value of $\tanh \left( 1 \right)$?
(a) 0
(b) 1
(c) 0.7616
(d) None of these

Answer 1

In this problem, we are trying to find the value of $\tanh \left( 1 \right)$. To start with, we will start by finding the values of $\sinh \left( 1 \right)$and $\cosh \left( 1 \right)$. Thus, putting the values back in the equation $\tanh \left( 1 \right)=\dfrac{\sinh \left( 1 \right)}{\cosh \left( 1 \right)}$, we will get a simplified answer in the form of a fraction including e. But, putting approximately 2.718 as e, we get our solution.

Complete step by step solution:
According to the problem, we are trying to find the value of $\tanh \left( 1 \right)$ .
To start with, we have, $\tanh x=\dfrac{\sinh x}{\cosh x}$ .
We also have, $\sinh x=\dfrac{{{e}^{x}}-{{e}^{-x}}}{2}$ and again, $\cosh x=\dfrac{{{e}^{x}}+{{e}^{-x}}}{2}$.
Now, we are trying to find the value of $\tanh \left( 1 \right)$.
So, to find it, we also have to find the values of $\sinh \left( 1 \right)$ and $\cosh \left( 1 \right)$.
Thus, to find the value of $\sinh \left( 1 \right)$, we have to put the value of x as 1.
Putting the value, we get, $\sinh \left( 1 \right)=\dfrac{{{e}^{1}}-{{e}^{-1}}}{2}$.
And again, we are trying to the value of $\cosh \left( 1 \right)$.
Thus, similarly, we also get, $\cosh \left( 1 \right)=\dfrac{{{e}^{1}}+{{e}^{-1}}}{2}$.
Then, we get, $\tanh \left( 1 \right)=\dfrac{\sinh \left( 1 \right)}{\cosh \left( 1 \right)}$.
Putting the values of $\sinh \left( 1 \right)$ and $\cosh \left( 1 \right)$ back, we are getting,
$\Rightarrow \tanh \left( 1 \right)=\dfrac{\dfrac{{{e}^{1}}-{{e}^{-1}}}{2}}{\dfrac{{{e}^{1}}+{{e}^{-1}}}{2}}$
Cancelling out now,
$\Rightarrow \tanh \left( 1 \right)=\dfrac{{{e}^{1}}-{{e}^{-1}}}{{{e}^{1}}+{{e}^{-1}}}$
From the rule of exponents, we also know, ${{e}^{-1}}=\dfrac{1}{e}$ ,
$\Rightarrow \tanh \left( 1 \right)=\dfrac{e-\dfrac{1}{e}}{e+\dfrac{1}{e}}$
After more simplification,
$\Rightarrow \tanh \left( 1 \right)=\dfrac{\dfrac{{{e}^{2}}-1}{e}}{\dfrac{{{e}^{2}}+1}{e}}$
Again, cancelling out, we are getting,
$\Rightarrow \tanh \left( 1 \right)=\dfrac{{{e}^{2}}-1}{{{e}^{2}}+1}$
We know that the value of e is approximately 2.718.
Putting the values and simplifying, we are getting,
$\Rightarrow \tanh \left( 1 \right)=0.7616$.

So, the correct answer is “Option c”.

Note: In this problem, we have used the value of e to get our solution. Euler’s Number ‘e’ is a numerical constant used in mathematical calculations. The value of e is 2.718281828459045…so on. Just like pi, e is also an irrational number.