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How to Put Cosh Inverse in Scientific Calculator

Reviewed by Calculator Editorial Team

The inverse hyperbolic cosine function, written as cosh⁻¹(x), is the inverse of the hyperbolic cosine function. It's used in various scientific and engineering applications, particularly in problems involving exponential growth or decay.

What is cosh⁻¹?

The hyperbolic cosine function, cosh(x), is defined as:

Formula

cosh(x) = (ex + e-x) / 2

The inverse hyperbolic cosine function, cosh⁻¹(x), is the value that when plugged into cosh(x) gives the original input x. It's defined for x ≥ 1 and has the formula:

Inverse Formula

cosh⁻¹(x) = ln(x + √(x² - 1))

This function is useful in physics, engineering, and mathematics for solving equations involving exponential growth or decay.

How to Calculate cosh⁻¹

To calculate the inverse hyperbolic cosine of a number, you can use the formula:

Calculation Steps

  1. Square the input value: x²
  2. Subtract 1 from the squared value: x² - 1
  3. Take the square root of the result: √(x² - 1)
  4. Add this to the original value: x + √(x² - 1)
  5. Take the natural logarithm (ln) of the sum

This gives you the value of cosh⁻¹(x).

Using a Scientific Calculator

Most scientific calculators have a dedicated function for inverse hyperbolic cosine. Here's how to use it:

  1. Turn on your scientific calculator
  2. Enter the value you want to calculate cosh⁻¹ for
  3. Look for the "cosh" or "cos⁻¹" button, often labeled with a "h" or "inv" symbol
  4. Press the "2nd" or "shift" function button (if required)
  5. Press the "cosh" button
  6. The calculator will display the result of cosh⁻¹(x)

Note

If your calculator doesn't have a direct cosh⁻¹ function, you can use the natural logarithm function (ln) with the formula shown above.

Worked Example

Let's calculate cosh⁻¹(2.5):

  1. Square the input: 2.5² = 6.25
  2. Subtract 1: 6.25 - 1 = 5.25
  3. Take the square root: √5.25 ≈ 2.2913
  4. Add to original value: 2.5 + 2.2913 ≈ 4.7913
  5. Take natural logarithm: ln(4.7913) ≈ 1.577

Therefore, cosh⁻¹(2.5) ≈ 1.577.

Common Mistakes

  • Trying to calculate cosh⁻¹ for values less than 1 - the function is only defined for x ≥ 1
  • Forgetting to use the natural logarithm (ln) instead of common logarithm (log)
  • Not pressing the shift or 2nd function before the cosh button on some calculators
  • Rounding intermediate values too early, which can lead to significant errors

FAQ

What is the domain of the inverse hyperbolic cosine function?

The domain of cosh⁻¹(x) is all real numbers x ≥ 1. This means the function is only defined for values of x that are 1 or greater.

How is inverse hyperbolic cosine different from inverse cosine?

The inverse hyperbolic cosine function (cosh⁻¹) is different from the inverse cosine function (cos⁻¹). While cos⁻¹ is defined for all real numbers between -1 and 1, cosh⁻¹ is only defined for x ≥ 1. The formulas and applications are also different.

Can I use a calculator to find cosh⁻¹ without a scientific mode?

If your calculator doesn't have a direct cosh⁻¹ function, you can use the natural logarithm function (ln) with the formula: cosh⁻¹(x) = ln(x + √(x² - 1)).