Question

Write three MATLAB functions to caiculate the hyperbolic sine, cosine, and tangent functions. $$ \sinh (x)=\frac{e^{x}-e^{-x}}{2}, \cosh (x)=\frac{e^{x}+e^{-x}}{2}, \tanh (x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} $$ Use your functions to plot the shapes of the hyperbolic sine, cosine, and tangent functions.

Short answer

Create three MATLAB functions to calculate the hyperbolic sine, cosine, and tangent functions as follows: 1. Hyperbolic Sine function: ```MATLAB function y = sinhfunc(x) y = (exp(x) - exp(-x))/2; end ``` 2. Hyperbolic Cosine function: ```MATLAB function y = coshfunc(x) y = (exp(x) + exp(-x))/2; end ``` 3. Hyperbolic Tangent function: ```MATLAB function y = tanhfunc(x) y = (exp(x) - exp(-x))/(exp(x) + exp(-x)); end ``` Then, plot the functions using this code: ```MATLAB x = linspace(-5, 5, 1000); y_sinh = sinhfunc(x); y_cosh = coshfunc(x); y_tanh = tanhfunc(x); subplot(3,1,1) plot(x, y_sinh) title('Hyperbolic Sine') subplot(3,1,2) plot(x, y_cosh) title('Hyperbolic Cosine') subplot(3,1,3) plot(x, y_tanh) title('Hyperbolic Tangent') ``` This will produce three separate plots for the hyperbolic sine, cosine, and tangent functions.

Step-by-step solution

1. Hyperbolic Sine function

Here is how to write a MATLAB function for the hyperbolic sine function: \[ \text{function y = sinhfunc(x)} \\ \text{y = (exp(x) - exp(-x))/2;} \\ \text{end} \]

2. Hyperbolic Cosine function

Here is how to write a MATLAB function for the hyperbolic cosine function: \[ \text{function y = coshfunc(x)} \\ \text{y = (exp(x) + exp(-x))/2;} \\ \text{end} \]

3. Hyperbolic Tangent function

And here is how to write a MATLAB function for the hyperbolic tangent function: \[ \text{function y = tanhfunc(x)} \\ \text{y = (exp(x) - exp(-x))/(exp(x) + exp(-x));} \\ \text{end} \]

4. Plotting the Functions

Now that we have the functions, we can make the plots of each function. Here's how to do it in MATLAB: \[ \text{x = linspace(-5, 5, 1000);} \\ \text{y_sinh = sinhfunc(x);} \\ \text{y_cosh = coshfunc(x);} \\ \text{y_tanh = tanhfunc(x);} \\ \text{subplot(3,1,1)} \\ \text{plot(x, y_sinh)} \\ \text{title('Hyperbolic Sine')} \\ \text{subplot(3,1,2)} \\ \text{plot(x, y_cosh)} \\ \text{title('Hyperbolic Cosine')} \\ \text{subplot(3,1,3)} \\ \text{plot(x, y_tanh)} \\ \text{title('Hyperbolic Tangent')} \\ \] This code will produce three separate plots, one each for the hyperbolic sine, cosine and tangent functions. The linspace function is used to generate a vector of evenly spaced points within a specified range, in this case from -5 to 5. This vector is then used as the input to each of the functions, and the output is plotted against the input. The subplot function is used to display multiple plots in a single figure window.

Key concepts

Hyperbolic Functions

Hyperbolic functions are mathematical functions that relate to the hyperbola, similar to how trigonometric functions relate to the circle. These functions are crucial in various fields such as engineering, physics, and hyperbolic geometry.
They are defined using exponential functions and are denoted by the prefixes "sinh," "cosh," and "tanh." These functions play a significant role in solving certain types of differential equations, describing the shape of hanging cables (catenary), and in the theory of special relativity.
In MATLAB, hyperbolic functions can be defined using the built-in exponential function combined in specific ways. These definitions rely heavily on the behavior of the exponential function itself. Let's explore each of these hyperbolic functions in detail.

Sinh Function

The sinh function, short for hyperbolic sine, is defined as the average difference of exponential functions. Mathematically, it is given by:
\[\sinh(x) = \frac{e^x - e^{-x}}{2}\]
This function is helpful in describing scenarios where growth trends occur symmetrically in both positive and negative directions. It's often used in physics, specifically in the calculations of special types of integrals and in hyperbolic geometry applications.
In MATLAB, the sinh function can be created with a simple function command:

• function y = sinhfunc(x)
• y = (exp(x) - exp(-x)) / 2;
• end

This code calculates the hyperbolic sine value for any input x.

Cosh Function

Cosh, or hyperbolic cosine, takes the average sum of exponential functions, forming a smooth curve that looks parabolic. Its mathematical definition is:
\[\cosh(x) = \frac{e^x + e^{-x}}{2}\]
The cosh function is often used in engineering, providing solutions for problems involving hyperbolic geometry and in determining the shape of cables under uniform gravitational force (known as the catenary curve).
Implementing a cosh function in MATLAB is straightforward:

• function y = coshfunc(x)
• y = (exp(x) + exp(-x)) / 2;
• end

This function computes the hyperbolic cosine for any given input.

Tanh Function

The tanh function, short for hyperbolic tangent, combines the hyperbolic sine and cosine functions to describe a curve that asymptotically approaches 1 and -1. Its mathematical formula is as follows:
\[\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}\]
This function is particularly useful in machine learning for activation functions in neural networks, as it helps normalize the output of functions.
In MATLAB, the tanh function can be programmed simply as:

• function y = tanhfunc(x)
• y = (exp(x) - exp(-x)) / (exp(x) + exp(-x));
• end

This code snippet enables the calculation of the hyperbolic tangent for a given input x. The output ranges between -1 and 1, which is why this function is often used for scaling values.