Understanding the Inverse of the Hyperbolic Cosine Function

Hyperbolic functions, such as the hyperbolic cosine (coshx), have their inverses, known as the inverse hyperbolic cosine or arcoshx. This article explores the formula and derivation for the inverse of the hyperbolic cosine function, as well as its properties.

The Formula for Inverse Hyperbolic Cosine

The inverse of the hyperbolic cosine function can be expressed using natural logarithms. The formula for this inverse function is:

arcoshx ln(x sqrt(x^2 - 1))

This formula is valid for x geq; 1, as the range of the hyperbolic cosine function is [1, infin;).

Deriving the Inverse Hyperbolic Cosine Function

To derive the inverse of the hyperbolic cosine function, we start with the definition of coshx:

coshx (e^x e^-x) / 2

We want to find a function that, when composed with coshx, returns x. Here’s how we can derive it step by step:

Step 1: Set Up the Equation and Solve for y

Let’s set y equal to coshx and solve for x:

y (e^x e^-x) / 2

Step 2: Solve for e^x

First, we clear the fraction by multiplying both sides by 2:

2y e^x e^-x

Next, let's isolate the terms involving e^x and e^-x:

2y e^x 1/e^x

Multiply both sides by e^x to clear the denominator:

2ye^x (e^x)^2 1

Now, we have a quadratic equation in terms of e^x:

(e^x)^2 - 2ye^x 1 0

Step 3: Apply the Quadratic Formula

Recall the quadratic formula x [-b plusmn; sqrt(b^2 - 4ac)] / 2a. In our equation, a 1, b -2y, and c 1. Plugging in these values, we get:

e^x [2y plusmn; sqrt(4y^2 - 4)] / 2

Factor out the 4 from the discriminant and simplify:

e^x [2y plusmn; 2sqrt(y^2 - 1)] / 2

Divide both the numerator and the denominator by 2:

e^x y plusmn; sqrt(y^2 - 1)

Step 4: Choose the Proper Root

Since coshx geq; 1, we need the positive branch, so we choose:

e^x y sqrt(y^2 - 1)

Finally, taking the natural logarithm of both sides, we have:

x ln(y sqrt(y^2 - 1))

This is the inverse hyperbolic cosine function, or:

arcoshy ln(y sqrt(y^2 - 1))

Properties and Usage of Inverse Hyperbolic Cosine

The inverse hyperbolic cosine function, arcoshx, has several important properties:

Domain: x 1 Codomain: [0, infin;) Monotonicity: arcoshx is strictly increasing. Reflection Property: arcosh(x) arcosh(y) arcosh(xy sqrt((x^2 - 1)(y^2 - 1))). Special Values: arcosh(1) 0, arcosh(e) 1.

The inverse hyperbolic cosine function is used in various fields of mathematics and physics, particularly in solving differential equations and in special relativity.

Conclusion

Understanding the inverse of the hyperbolic cosine function is crucial for advanced mathematical and physical applications. The formula and derivation presented here provide a comprehensive guide to working with this function. If you found this article helpful, please consider sharing it.