Adventures in Machine Learning

Mastering Hyperbolic Trigonometric Functions: Formulas Implementation and Plotting

Hyperbolic Trigonometric Function: What You Need to Know About These Special Functions

Do you love mathematics and want to explore some advanced concepts? Are you puzzled about the hyperbolic trigonometric functions and wondering about their significance?

This article will provide you with a comprehensive overview of hyperbolic trigonometric functions, including their definition, formulas, and implementation. By the end of this article, you will have a clear understanding of hyperbolic functions and how they differ from standard trigonometric functions.

Definition of Hyperbolic Trigonometric Function

Hyperbolic trigonometric functions are a set of six functions defined based on the hyperbola instead of the circle. The six functions are hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant and hyperbolic cosecant.

These functions are commonly abbreviated as sinh, cosh, tanh, coth, sech, and csch, respectively.

Difference between normal trigonometric functions and hyperbolic functions

The main difference between hyperbolic and circular trigonometry is their definition. Circular trigonometry deals with angles and relationships between points on the unit circle, while hyperbolic trigonometry deals with hyperbolas and relationships between points on the hyperbola.

Moreover, the range of the hyperbolic functions is different from circular functions.

Formulas of Hyperbolic Functions

The formulas of the hyperbolic functions are related to the exponential function, and they are expressed using exponential functions such as

sinh(x) = (e^ x - e^-x)/2
cosh(x) = (e^ x + e^-x)/2
tanh(x) = (e^ x - e^-x)/(e^ x + e^-x)
coth(x) = (e^ x + e^-x)/(e^ x - e^-x)
sech(x) = 1 / cosh(x)
csch(x) = 1 / sinh(x)

Hyperbolic Cosine Function

The hyperbolic cosine function, denoted as cosh(x), is a fundamental function in the concept of the hyperbolic trigonometric function. It is defined as the ratio of the adjacent side to the hypotenuse in a right triangle with one acute angle equal to x (radians), where the opposite angle and the hypotenuse connect with two points on the hyperbola.

The domain and range of the hyperbolic cosine function are all real numbers. The shape of the hyperbolic cosine function is a “U-like” shape, where the minimum is at zero.

Implementation of Hyperbolic Cosine Function using Numpy.cosh()

Numpy is a well-known and frequently used Python library for scientific programming. The Numpy library provides a built-in function named cosh() to evaluate the hyperbolic cosine function.

The numpy.cosh() function is computed using the formula: 

cosh(x) = (e^x + e^-x)/2.

The syntax and parameters of Numpy.cosh() function

numpy.cosh( x, / , out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])
  • x: Input values expressed in radians.
  • out: Optional, output array.
  • where: Optional, this parameter allows you to specify an alternative result if the condition is False.
  • casting: Controls what automatic casting rules apply.
  • order: Controls the memory layout order of the result.
  • dtype: Controls the data type of the output.
  • subok: Set to True by default, to allow subclasses.

Four Methods to implement Hyperbolic Cosine Function

  1. Simple and direct calculation: Using the formula cosh(x) = (e^x + e^-x)/2 to evaluate the hyperbolic cosine function without any third-party libraries.
  2. Using Pythons math module: The math module provides built-in functions to evaluate different functions like sine, cosine, and tangent. The users can evaluate the hyperbolic cosine function using the cosh() function provided by the math module.
  3. Using Numpy Package: Numpy provides the cosh(), which focuses only on calculating the hyperbolic cosine function.
  4. Using Scipy Package: Scipy is an open-source library that provides various mathematical algorithms that cover optimization, integration, interpolation, and many more.
  5. Scipy provides a hyperbolic cosine function named cosh() to evaluate the hyperbolic cosine.

Conclusion

This article explains the fundamentals of hyperbolic trigonometric functions along with the formulas and implementation of the hyperbolic cosine function. We discussed the difference between standard trigonometric functions and hyperbolic functions.

Finally, we introduced the four methods to evaluate the hyperbolic cosine function. We hope that this article will help spark your interest in the subject and that you’ll continue to explore these advanced mathematical concepts in the future.

Passing Single and Multi-valued Parameter in Hyperbolic Trigonometric Function

The hyperbolic cosine function is a fundamental function in the concept of hyperbolic trigonometric functions. It is defined as the ratio of the adjacent side to the hypotenuse in a right triangle with one acute angle equal to x (radians).

In Python, the hyperbolic cosine function can be evaluated using different techniques, including passing parameters. In this article, we will be discussing the implementation of the hyperbolic cosine function with single- and multi-valued parameters.

Implementation of Hyperbolic Cosine Function with a Single-valued Parameter

In Python, you can evaluate the hyperbolic cosine function using the cosh() method provided by various libraries, such as Numpy and Math module. The Numpy library provides a built-in function named cosh() to evaluate the hyperbolic cosine function.

With the Numpy library, passing a single-valued parameter to the cosh() function is relatively simple. The syntax for passing a single-valued parameter to Numpy.cosh() function is as follows: 

numpy.cosh(x)

Here, x is a single-valued parameter that specifies the angle in radians.

For example, to evaluate the hyperbolic cosine of the angle 2, we can pass it as a single-valued parameter as follows:

import numpy as np
x = 2
result = np.cosh(x)
print(result)

Output:

3.762195691080089

Thus, by passing a single-valued parameter to the numpy.cosh() function, we can evaluate the hyperbolic cosine function.

Implementation of Hyperbolic Cosine Function with Multi-valued Parameter

In addition to single-valued parameters, we can also pass multi-valued parameters to the hyperbolic cosine function.

Multi-valued parameters are used while we need to calculate the cosh() function on multiple values. Here, we pass an array or a list of angles to the cosh() function, and it returns an array of the same shape as the input array containing the results.

The syntax for passing multi-valued parameters to Numpy.cosh() is as follows:

numpy.cosh(x, / , out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

Here, the first parameter x is the input array providing an angle or array of angles in radians. For example, consider the following code to calculate the hyperbolic cosine function on an array of angles:

import numpy as np
x = np.array([0, 1, 2, 3])
result = np.cosh(x)
print(result)

Output:

[ 1.          1.54308063  3.76219569 10.06766196]

Thus, by passing multi-valued parameters to the numpy.cosh() function, we can calculate the hyperbolic cosine function for an array of angles.

Providing an Optional Output Variable

While evaluating complex calculations, we may need to store our results into an already defined array instead of creating a new one. In the case of numpy.cosh() function, we have an optional output variable, which allows us to define an output array as an argument.

Explanation of the “out” Parameter in Numpy.cosh() Function

The “out” parameter in Numpy.cosh() function is an optional parameter that allows us to define an output array to store the result from numpy.cosh() function. It is used when we do not want to allocate a new array for calculation but to reuse a pre-allocated array instead.

The syntax for the “out” parameter of Numpy.cosh() function is as follows:

numpy.cosh(x, / , out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

Here, the second parameter “out” is an optional parameter that holds the output array. By default, out parameter is None which means a new array will be allocated for the result, but we can reuse a pre-allocated array in place of new allocation by passing that array to the out parameter.

Implementation of Hyperbolic Cosine Function with an Optional Output Variable

The following example demonstrates how to use the “out” parameter while determining the hyperbolic cosine function:

import numpy as np
x = np.array([0, 1, 2, 3])
out_array = np.zeros(4)
np.cosh(x, out=out_array)
print(out_array)

Output:

[ 1.          1.54308063  3.76219569 10.06766196]

In the above example, we initialized an output array of size 4 with zero values.

In the np.cosh() function, we pass the output array to the out parameter. After the function is completed, the values of the hyperbolic cosine function are stored in the output array.

Conclusion

In conclusion, we have discussed the implementation of the hyperbolic cosine function using single- and multi-valued parameters, along with the optional output variable. Passing single-valued parameters is used when we need to calculate the cosh() function for a single angle.

In contrast, multi-valued parameters are used to generate cosh() function values for an array of angles. The “out” parameter is optional but helpful when we need to store the result in a pre-allocated array.

By understanding and using these methods, you can use the hyperbolic cosine function in your Python programs with ease.

Passing a Complex Variable and Plotting Numpy.cosh() on a Graph

In Python, the hyperbolic cosine function can be evaluated not just for real values but also for complex values.

This makes it possible to use the hyperbolic cosine function in a wide range of mathematical applications. Additionally, we can use the Matplotlib library to plot the hyperbolic cosine function and visualize its behavior in a graph.

In this article, we will be discussing how to implement the hyperbolic cosine function with a complex variable and how to plot it on a graph using the Matplotlib library.

Implementation of Hyperbolic Cosine Function with a Complex Variable

The Numpy library provides the cosh() function, which accepts a complex variable as input.

The complex variable can be represented as a+bi, where a and b are real numbers and i is the imaginary unit. The hyperbolic cosine function will return a complex number as output.

The formula for evaluating the hyperbolic cosine function with a complex variable is:

cosh(z) = (e^z + e^-z)/2

Here, z is a complex value. To implement the hyperbolic cosine function with a complex variable, we can create a complex variable using the complex() method provided by Python and pass it to the cosh() function provided by the Numpy library.

The syntax for evaluating the hyperbolic cosine function with a complex variable using Numpy is as follows:

import numpy as np
z = complex(a, b)
result = np.cosh(z)
print(result)

Here, the complex() method creates a complex variable z with real part a and imaginary part b. The cosh() function then calculates the hyperbolic cosine function for the complex variable z and stores it in the variable result.

Plotting Numpy.cosh() on a Graph

Now that we know how to evaluate the hyperbolic cosine function with complex variables, let us learn how to plot the function on a graph using the Matplotlib library. The Matplotlib library provides numerous tools for creating visualizations in Python.

We can use it to plot the hyperbolic cosine function by first creating an array of input values and an array of corresponding output values and then using the plot() method provided by the Matplotlib library.

Importing Matplotlib Library to Plot the Function

The first step is to import the Matplotlib library. This can be done using the following code:

import matplotlib.pyplot as plt

This imports the entire Matplotlib library and renames it as plt, making it easier to use in our code.

Representation of Hyperbolic Cosine Function using Input and Output Array

To plot the hyperbolic cosine function, we first need to create an array of input values and an array of corresponding output values. We can use the Numpy library to create these arrays.

The following code snippet creates an array of input values from -5 to 5 with a step size of 0.1:

import numpy as np
x = np.arange(-5, 5, 0.1)

Next, we can use the cosh() function provided by the Numpy library to generate an array of output values using the input values array:

y = np.cosh(x)

Now that we have the input and output arrays, we can use the plot() method provided by the Matplotlib library to represent the hyperbolic cosine function on a graph. The following code snippet does this:

import matplotlib.pyplot as plt
import numpy as np
x = np.arange(-5, 5, 0.1)
y = np.cosh(x)
plt.plot(x, y)
plt.xlabel('Input Values')
plt.ylabel('Output Values')
plt.title('Hyperbolic Cosine Function')
plt.show()

The plot() method takes in two arrays as arguments: the input array, represented by x, and the output array, represented by y. The xlabel(), ylabel(), and title() methods are used to add a label to the x-axis, the y-axis, and the plot, respectively.

The show() method is used to display the plot on the screen.

Conclusion

In conclusion, we have learned how to implement the hyperbolic cosine function with a complex variable and how to plot the function on a graph using the Matplotlib library. Evaluating the hyperbolic cosine function with complex variables opens up the function to a wide range of mathematical applications.

While plotting functions using Matplotlib library provides a visual representation of the function’s behavior that can be useful in various scientific and engineering fields. By understanding and utilizing these techniques, you can enhance your knowledge of hyperbolic functions and apply them to real-world problems.

Summary

The hyperbolic cosine function is a fundamental function in the concept of hyperbolic trigonometric functions. In this article, we have discussed the basics of the hyperbolic cosine function, including its definition, formulas, and implementation.

We started by introducing hyperbolic trigonometric functions as a set of six functions defined based on the hyperbola instead of the circle. Unlike circular trigonometry, hyperbolic trigonometry deals with hyperbolas and relationships between points on the hyperbola.

We then discussed the difference between hyperbolic and standard trigonometric functions. We then delved into the specifics of the hyperbolic cosine function, which is defined as the ratio of the adjacent side to the hypotenuse in a right triangle with one acute angle equal to x (radians), where the opposite angle and hypotenuse connect with two points on the hyperbola.

With implementations using Numpy, Math module, and Scipy packages, we learned how to evaluate the hyperbolic cosine function with single- and multi-valued parameters along

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