Tanh
You have already learned about trigonometric functions and their properties. Unlike trigonometric functions, hyperbolic functions are defined for the hyperbola.
In this short lesson, you will learn about Tanh hyperbolic function and its properties including tanh function, tanh calculator, tanh formula, and tanh equation.
You can see the graphical representation of the Tanh function in the image below.
Lesson Plan
What Is Meant by Tanh?
Tanh is a hyperbolic function that is pronounced as "tansh."
The function Tanh is the ratio of Sinh and Cosh.
\[\tanh=\dfrac{\sinh}{\cosh}\]
We can even work out with exponential function to define this function.
\[\tanh=\dfrac{e^{x}e^{x}}{e^{x}+e^{x}}\]
What Are the Formulas for Tanh Function?
There is more than one formula for the Tanh function.
Here are a few of the Tanh formulas.
\[\tanh{(x)}=\tanh{(x)}\] 
\[\tanh{(x \pm y)}=\dfrac{\tanh{(x)} \pm \tanh{(y)}}{1 \pm \tanh{(x)}\tanh{(y)}}\] 
\[\tanh{(2x)}=\dfrac{2\tanh{(x)}}{1 + \tanh^{2}{(x)}}\] 
\[\tanh{\dfrac{x}{2}}=\dfrac{\sinh{x}}{1 + \cosh{x}}=\dfrac{\cosh{x}1}{\sinh{x}}\] 
Graphical Representation of Tanh Function
You can see the graphical representation of the Tanh function in the image above.
Notice that the graph is symmetric along the origin.
This means that the function is odd.
Here is a Tanh calculator for you to see a graphical representation of all hyperbolic functions.
What Are the Properties of Tan Function?
The word "tan" stands for the tangent.
The tangent function "\(\tan{x}\)" is one of the six basic trigonometric functions and its formula is:
\(\tan{x}=\dfrac{\sin{x}}{\cos{x}}\) 
Properties of \(\tan{x}\)
 Negative value of the angle gives negative value of the tan function, \(\tan{x}=\tan{x}\)
 It is a periodic function and its period is \(\pi\)
 It is symmetric about the origin.
 Its domain is a set of all real values except \(x=\dfrac{\pi}{2}+n\pi\), where \(n\) is an integer.
 Its range is \((\infty,\infty)\).
The properties of the hyperbolic functions are corresponding to the trigonometric functions.
Tanh equation that gives the relation between tan function and tanh function is:
\(\tanh{x}=i\tan{ix}\), where \(i=\sqrt{1}\) 
Example
Calculate the value of \(\tan{\left(\dfrac{\pi}{4}\right)}\) by using properties of tan function.
To calculate it, apply the property \(\tan{x}=\tan{x}\).
\[\begin{aligned}\tan{\left(\dfrac{\pi}{4}\right)}&=\tan{\left(\dfrac{\pi}{4}\right)}\\&=1\end{aligned}\]
What Is the Inverse of Tan Function?
The inverse of tan function is represented as \(\tan^{1}{x}\).
The inverse function \(\tan^{1}\) is also called \(a\tan\) and \(arc\tan\).
The inverse function \(\tan^{1}{x}\) gives the value of the angle \(\theta\).
\[\begin{aligned}\tan{\theta}&=x\\\theta&=\tan^{1}{x}\end{aligned}\] 

Tanh is the ratio of sinh and cosh.

Tanh function is defined for all real numbers.

The range of Tanh function is \((1, 1)\).

Tanh satisfies \(\tanh{(x)}=\tanh{(x)}\); so it is an odd function.
Solved Examples
Example 1 
We know that \(\tanh=\dfrac{\sinh}{\cosh}\).
Use the representation of \(\sinh\) and \(\cosh\) in terms of exponential function to derive the formula \(\tanh=\dfrac{e^{x}e^{x}}{e^{x}+e^{x}}\).
Solution
The hyperbolic function \(\sinh{x}\) is given by:
\[\sinh{x}=\dfrac{e^{x}e^{x}}{2}\]
The hyperbolic function \(\cosh{x}\) is given by:
\[\cosh{x}=\dfrac{e^{x}+e^{x}}{2}\]
\(tanh\) is the ratio of \(\sinh{x}\) and \(\cosh{x}\).
\[\begin{align}\tanh{x}&=\dfrac{\sinh{x}}{\cosh{x}}\\\tanh{x}&=\left(\dfrac{e^{x}e^{x}}{2}\right) \div \left(\dfrac{e^{x}+e^{x}}{2}\right)\\\tanh{x}&=\left(\dfrac{e^{x}e^{x}}{2}\right) \times\left(\dfrac{2}{e^{x}+e^{x}}\right)\\\tanh&=\dfrac{e^{x}e^{x}}{e^{x}+e^{x}}\end{align}\]
Hence, proved. 
Example 2 
What is the relation between \(\tan\) function and \(\tanh\) function?
Can you prove it?
Solution
The relation between \(\tan\) function and \(\tanh\) function is \(\tan{x}=i\tanh{ix}\).
Proof:
The hyperbolic function \(\sinh{x}\) is given by:
\[\sinh{x}=\dfrac{e^{x}e^{x}}{2}\]
Now substitute \(ix\) for \(x\) in the above equation and apply the hyperbolic property \(\sinh{ix}=i\sin{x}\).
\[\begin{aligned}\sinh{ix}&=\dfrac{e^{ix}e^{ix}}{2}\\i\cdot\sin{x}&=\dfrac{e^{ix}e^{ix}}{2}\\\sin{x}&=\dfrac{e^{ix}e^{ix}}{2i}\cdots(1)\end{aligned}\]
The hyperbolic function \(\cosh{x}\) is given by:
\[\cosh{x}=\dfrac{e^{x}+e^{x}}{2}\]
Now substitute \(ix\) for \(x\) in the above equation and apply the hyperbolic property \(\cosh{ix}=i\cos{x}\).
\[\begin{aligned}\cosh{ix}&=\dfrac{e^{ix}+e^{ix}}{2}\\\cos{x}&=\dfrac{e^{ix}+e^{ix}}{2}\cdots(2)\end{aligned}\]
Now divide equation (1) by equation (2).
\[\begin{aligned}\dfrac{\sin{x}}{\cos{x}}&=\dfrac{\dfrac{e^{ix}e^{ix}}{2i}}{\dfrac{e^{ix}+e^{ix}}{2}}\\\tan{x}&=\dfrac{1}{i}\cdot\dfrac{e^{ix}e^{ix}}{e^{ix}+e^{ix}}\\&=\dfrac{\tanh{ix}}{i}\\&=\dfrac{i\tanh{ix}}{i\cdot i}\\&=\dfrac{i\tanh{ix}}{i^2}\\&=i\tanh{ix}\end{aligned}\]
\(\therefore\) \(\tan{x}=i\tanh{ix}\) 
Example 3 
The height of a building is 200 feet.
The building is at a distance of 150 feet from point A.
Can you calculate the angle \(\theta\) by using the \(arctan\) rule?
Solution
The triangle formed is a rightangle triangle.
Now apply the inverse of the tangent equation and calculate the value of angle \(theta\) using a calculator.
\[\begin{align}\theta&=\tan^{1}\left(\dfrac{\text{Perpendicular}}{\text{Base}}\right)\\&=\tan^{1}\left(\dfrac{200}{150}\right)\\&=\tan^{1}\left(\dfrac{4}{3}\right)\\&=53.13^{\circ}\end{align}\]
\(\therefore\) The angle is \(\theta=53.13^{\circ}\) 
 Can you draw the graph of hyperbolic function \(\coth{x}\) by using the graph of \(\tanh{x}\)?
 Calculate the relation between \(\coth\) function and \(\cot\) function by using the tan hyperbolic property \(tanx=i\tanh{ix}\).
Interactive Questions
Here are a few activities for you to practice.
Select/type your answer and click the "Check Answer" button to see the result.
Let's Summarize
This minilesson targeted the fascinating concept of Tanh. The math journey around Tanh starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp but will also stay with them forever. Here lies the magic with Cuemath.
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Frequently Asked Questions (FAQs)
1. Is Tanh inverse Tan?
No, Tanh is not the inverse of Tan.
2. What is the range of Tanh?
The range of Tanh function is (1, 1).
3. What does Sine mean?
Sine is a trigonometric function.
In a rightangled triangle, it is the ratio of perpendicular to the hypotenuse.