arc tanh

What is arc tanh in math? Definition

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.

History of arc tanh

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.

What grade level is arc tanh for?

Arc tanh is typically introduced in advanced high school or college-level mathematics courses, such as calculus or advanced algebra.

What knowledge points does arc tanh contain? And detailed explanation step by step

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:

1. Start with a given value of x, which should be between -1 and 1.
2. Use the formula arctanh(x) = (1/2) * ln((1+x)/(1-x)), where ln represents the natural logarithm.
3. Substitute the value of x into the formula and simplify the expression.
4. Calculate the natural logarithm of the resulting expression.
5. Multiply the result by 1/2 to obtain the final value of arc tanh.

Types of 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.

Properties of arc tanh

The arc tanh function has several properties that are similar to other inverse trigonometric functions. Some of the notable properties include:

1. The domain of arc tanh is the set of real numbers between -1 and 1, inclusive.
2. The range of arc tanh is the set of all real numbers.
3. The function is odd, meaning arctanh(-x) = -arctanh(x).
4. The function is increasing, meaning if x1 < x2, then arctanh(x1) < arctanh(x2).

How to find or calculate arc tanh?

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.

What is the formula or equation for arc tanh?

The formula for arc tanh is arctanh(x) = (1/2) * ln((1+x)/(1-x)), where ln represents the natural logarithm.

How to apply the arc tanh formula or equation?

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.

What is the symbol or abbreviation for arc tanh?

The symbol or abbreviation for arc tanh is arctanh(x) or tanh^(-1)(x).

What are the methods for arc tanh?

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.

More than 3 solved examples on arc tanh

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(∞) = ∞.

Practice Problems on arc tanh

1. Calculate arctanh(0).
2. Find the value of arctanh(0.2).
3. Determine the arc tanh of -1.

FAQ on arc tanh

Question: What is the range of arc tanh? Answer: The range of arc tanh is the set of all real numbers.