The arc tanh, also known as the inverse hyperbolic tangent, is a mathematical function that is the inverse of the hyperbolic tangent function. It is denoted as arctanh(x) or tanh^(-1)(x), where x is a real number.
The concept of hyperbolic functions, including the hyperbolic tangent, dates back to the 18th century. However, the specific term "arc tanh" was coined later to represent the inverse of the hyperbolic tangent function.
Arc tanh is typically introduced in advanced high school or college-level mathematics courses, such as calculus or advanced algebra.
Arc tanh involves several key concepts in mathematics, including trigonometry, calculus, and algebra. Here is a step-by-step explanation of how to calculate arc tanh:
Arc tanh is a single-valued function, meaning it has a unique output for each input within its domain. It does not have multiple branches or variations like some other trigonometric or inverse trigonometric functions.
The arc tanh function has several properties that are similar to other inverse trigonometric functions. Some of the notable properties include:
To find or calculate the value of arc tanh, you can use a scientific calculator or computer software that has the function built-in. Alternatively, you can use the formula mentioned earlier to manually calculate the value.
The formula for arc tanh is arctanh(x) = (1/2) * ln((1+x)/(1-x)), where ln represents the natural logarithm.
To apply the arc tanh formula, substitute the value of x into the formula and simplify the expression. Then, calculate the natural logarithm of the resulting expression and multiply the result by 1/2 to obtain the final value of arc tanh.
The symbol or abbreviation for arc tanh is arctanh(x) or tanh^(-1)(x).
The main method for calculating arc tanh is using the formula mentioned earlier. However, as mentioned before, you can also use a scientific calculator or computer software that has the function built-in.
Example 1: Find the value of arctanh(0.5). Solution: Using the formula, arctanh(0.5) = (1/2) * ln((1+0.5)/(1-0.5)) = (1/2) * ln(1.5/0.5) = (1/2) * ln(3) ≈ 0.5493.
Example 2: Calculate arctanh(-0.8). Solution: Using the formula, arctanh(-0.8) = (1/2) * ln((1+(-0.8))/(1-(-0.8))) = (1/2) * ln(0.2/1.8) = (1/2) * ln(1/9) ≈ -1.0986.
Example 3: Determine the value of arctanh(1). Solution: Using the formula, arctanh(1) = (1/2) * ln((1+1)/(1-1)) = (1/2) * ln(2/0) = (1/2) * ln(∞) = ∞.
Question: What is the range of arc tanh? Answer: The range of arc tanh is the set of all real numbers.