In mathematics, tanh stands for hyperbolic tangent. It is a mathematical function that is widely used in various fields, including calculus, statistics, and physics. Tanh is a hyperbolic function that relates to the hyperbolic sine (sinh) and hyperbolic cosine (cosh) functions.
The concept of hyperbolic functions, including tanh, was first introduced by Swiss mathematician Leonhard Euler in the 18th century. These functions were initially developed to solve problems related to the geometry of hyperbolas. Over time, their applications expanded to various branches of mathematics and other scientific disciplines.
Tanh is typically introduced in advanced high school mathematics or early college-level courses. It is commonly covered in courses such as precalculus, calculus, and differential equations.
Tanh is a function that maps real numbers to values between -1 and 1. It can be defined using the exponential function as follows:
tanh(x) = (e^x - e^(-x)) / (e^x + e^(-x))
To calculate the value of tanh(x), follow these steps:
Tanh is a single-valued function, meaning it has a unique output for each input. It is continuous and differentiable for all real numbers.
Some important properties of tanh include:
To find or calculate the value of tanh(x), you can use scientific calculators or computer software that have built-in functions for hyperbolic tangent. Alternatively, you can use the formula mentioned earlier and perform the calculations manually.
The formula for tanh is:
tanh(x) = (e^x - e^(-x)) / (e^x + e^(-x))
To apply the tanh formula, substitute the value of x into the equation and perform the necessary calculations to obtain the value of tanh(x).
The symbol or abbreviation for tanh is "tanh".
The most common method for calculating tanh is by using the formula mentioned earlier. However, as mentioned before, scientific calculators and computer software also provide built-in functions for calculating tanh.
Example 1: Calculate tanh(2)
Using the formula, we have:
tanh(2) = (e^2 - e^(-2)) / (e^2 + e^(-2))
Calculating the exponential terms:
e^2 ≈ 7.389 e^(-2) ≈ 0.135
Substituting the values:
tanh(2) ≈ (7.389 - 0.135) / (7.389 + 0.135) ≈ 7.254 / 7.524 ≈ 0.964
Example 2: Calculate tanh(0)
Using the formula, we have:
tanh(0) = (e^0 - e^(-0)) / (e^0 + e^(-0))
Since e^0 = 1 and e^(-0) = 1, the equation simplifies to:
tanh(0) = (1 - 1) / (1 + 1) = 0 / 2 = 0
Example 3: Calculate tanh(-1)
Using the formula, we have:
tanh(-1) = (e^(-1) - e^(1)) / (e^(-1) + e^(1))
Calculating the exponential terms:
e^(-1) ≈ 0.368 e^(1) ≈ 2.718
Substituting the values:
tanh(-1) ≈ (0.368 - 2.718) / (0.368 + 2.718) ≈ -2.35 / 3.086 ≈ -0.76
Question: What is tanh? Answer: Tanh is a hyperbolic tangent function that maps real numbers to values between -1 and 1. It is widely used in mathematics and various scientific fields.