Problem 6 - Essential Calculus

Question

Evaluate the function. If the value is not a rational number, round your answer to three decimal places. (a) \(\operatorname{csch}^{-1} 2\) (b) \(\operatorname{coth}^{-1} 3\)

Step-by-Step Solution

Answer

\(\operatorname{csch}^{-1}(2) \approx 0.8814\) and \(\operatorname{coth}^{-1}(3) = \ln {\sqrt{2}} \approx 0.347\)

Step 1: Compute \(\operatorname{csch}^{-1}(2)\)

To compute the inverse hyperbolic cosecant of 2, you could refer to a calculator with this function - it should provide an answer. Alternatively, you could use the formula \(\operatorname{csch}^{-1}(x) = \ln \left(\frac{1}{x} + \sqrt{1 + \frac{1}{x^2}}\right)\), so computing \(\operatorname{csch}^{-1}(2)\), we obtain: \(\ln \left(\frac{1}{2} + \sqrt{1 + \frac{1}{4}}\right) = \ln{(1.1180)}\).

Step 2: Compute \(\operatorname{coth}^{-1}(3)\)

To compute the inverse hyperbolic cotangent of 3, again, a calculator that includes this function can help. Otherwise, you can use the formula \(\operatorname{coth}^{-1} (x) = \frac{1}{2}\ln\left(\frac{x + 1}{x - 1}\right)\). So \(\operatorname{coth}^{-1}(3) = \frac{1}{2}\ln \left(\frac{4}{2}\right) = \frac{1}{2}\ln{(2)} = \ln {\sqrt{2}}\).

Key Concepts

csch inverse function

The inverse hyperbolic cosecant function, denoted as \( \operatorname{csch}^{-1}(x) \), is a fascinating aspect of hyperbolic functions. It helps us find the angle whose hyperbolic cosecant is a given number. This might sound a bit complex at first, but it becomes simple with a formula! For anyone trying to compute \( \operatorname{csch}^{-1}(x) \), the formula \( \operatorname{csch}^{-1}(x) = \ln \left(\frac{1}{x} + \sqrt{1 + \frac{1}{x^2}}\right) \) is really handy.
In the context of the exercise, when you want to find \( \operatorname{csch}^{-1}(2) \), just plug 2 into this formula. This leads to:
\[ \operatorname{csch}^{-1}(2) = \ln \left(\frac{1}{2} + \sqrt{1 + \frac{1}{4}}\right) = \ln(1.1180) \]
This value, \( \ln(1.1180) \), gives you a clear understanding of the inverse hyperbolic cosecant of 2. Remember, it's all about translating the ratio back into an angle measure in the hyperbolic sense.

• In hyperbolic functions context, inverse operations calculate the original angle or argument.
• The output of \( \operatorname{csch}^{-1} \) serves as solving ripple effects in physics and engineering contexts.

coth inverse function

The inverse hyperbolic cotangent, \( \operatorname{coth}^{-1}(x) \), is another intriguing hyperbolic function. This function helps determine the hyperbolic angle whose cotangent is a particular value. Sounds challenging? Let's break it down with the formula \( \operatorname{coth}^{-1}(x) = \frac{1}{2}\ln\left(\frac{x + 1}{x - 1}\right) \).
When computing \( \operatorname{coth}^{-1}(3) \) from the problem, the formula comes to the rescue:
\[ \operatorname{coth}^{-1}(3) = \frac{1}{2}\ln \left(\frac{4}{2}\right) = \frac{1}{2}\ln(2) = \ln(\sqrt{2}) \]
By applying this formula, you find that \( \ln(\sqrt{2}) \) provides the solution. Keep in mind that the inverse of hyperbolic functions like this one plays a crucial role in different mathematical and computational fields, offering deeper insights into solving complex problems.

• These functions ensure you understand angles in hyperbolic geometry.
• Real-world applications often require reverse calculations using these inverse formulas.

hyperbolic functions

Hyperbolic functions are a set of six functions defined similarly to the trigonometric functions but for hyperbolic angles.
Common hyperbolic functions include:

• Hyperbolic sine (\( \sinh \))
• Hyperbolic cosine (\( \cosh \))
• Hyperbolic tangent (\( \tanh \))
• Hyperbolic cosecant (\( \operatorname{csch} \))
• Hyperbolic secant (\( \operatorname{sech} \))
• Hyperbolic cotangent (\( \operatorname{coth} \))

These functions resemble exponential forms in nature. If you've ever explored the walkway on a suspension bridge or observed the pattern of a cable, you're directly seeing hyperbolic shapes in action. For hyperbolic functions, they're crucial when modeling many physical phenomena, such as:

• The behavior of particles in relativistic physics.
• The shape of a hanging cable or chain, known as a catenary in architecture.

When delving deeper into hyperbolic functions, you'll realize that just like their trigonometric counterparts, they also have inverse roles. These inverses solve equations involving hyperbolic functions, unraveling the needed value you started with, which can be encountered in various scientific and mathematical contexts.