What are Hyperbolic Functions, their Use & Derivatives

Hyperbolic functions are as useful in mathematics as the trigonometric functions, the logarithmic functions, or the exponential functions. These functions, such as the hyperbolic sine (sinh) and the hyperbolic cosine (cosh), are defined using exponential functions, and they arise in Calculus, Complex Analysis, Physics and Engineering. From differential equations to space geometry, hyperbolic functions are important when solving such problems and errors.

This article discusses hyperbolic functions and their derivatives and describes how to do hyperbolic functions on TI 84 calculator.

What are Hyperbolic Functions?

Hyperbolic functions are similar to trigonometric functions, but instead of using circles, they involve hyperbolas. It is possible to define them by exponential functions, and they are used in calculus, complex analysis, and hyperbolic geometry.

The primary hyperbolic functions are:

1. Hyperbolic Sine (sinh):

 sinh(x) = {ex – e-x}÷{2}

2. Hyperbolic Cosine (cosh):

 cosh(x) = {ex+ e-x}÷{2}

3. Hyperbolic Tangent (tanh):

tanh(x) = {sinh(x)}÷{cosh(x)} = {ex – e-x}÷{ex + e-x}

4. Hyperbolic Cotangent (coth):

coth(x) = {cosh(x)}÷{sinh(x)} = {ex + e-x}÷{ex – e-x}

5. Hyperbolic Secant (sech):

sech(x) = {1}÷{cosh(x)} = {2}÷ {ex + e-x}

6. Hyperbolic Cosecant (csch):

csch(x) ={1}÷{sinh(x)} = {2}÷{ex – e-x}

How to Use Hyperbolic Functions on a TI 84 Calculator?

With the built-in capabilities of the TI-84 calculator, one can solve hyperbolic functions. Here’s a step-by-step guide:

1. Turn on the calculator

2. Access the Hyperbolic Functions

• First, “Go to your calculator and press the “MATHS” button.”
• Navigate to the “NUM” tab by gently pressing the right arrow key on the keyboard until the arrow reaches the “NUM” tab.
• Scroll down this page to see some of the important hyperbolic functions, including sinh, cosh, and tanh.

3. Compute a Hyperbolic Function

• Choose the desired function
• Enter the value of x.
• Press “Enter” to see the result.

For example, to compute sinh⁡(1)

• Press “MATH”.
• Scroll right to “NUM.”
• Select 3: sinh(.
• Enter 1.
• Press “ENTER”.

The screen should display the result of sinh⁡(1).

What are the Derivatives of Hyperbolic Functions?

Here are the derivatives of the primary hyperbolic functions:

Function	Derivative
sinh⁡(𝑥)	cosh⁡(𝑥)
cosh⁡(𝑥)	sinh⁡(𝑥)
tanh⁡(𝑥)	\sech2(𝑥)
coth⁡(𝑥)	−\csch2(𝑥)
\sech(𝑥)	−\sech(𝑥)tanh⁡(𝑥)
\csch(𝑥)	−\csch(𝑥)coth⁡(𝑥)

Conclusion

Hyperbolic functions are trigonometric functions that are used in hyperbolas and work well in theoretical mathematics as well as in practical applications. As for computing these functions, it is relatively easy on the TI-84 calculator. On the level of derivatives, these functions suggest further applications in calculus and analysis.

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