## Introduction to Arctan and NumPy Arctan

Arctan, also known as inverse tangent, is a trigonometric function that is used to determine the angle whose tangent is equal to a given number. In mathematical terms, if tan θ = y then θ = arctan(y).

The range of arctan is between -π/2 to π/2 and it is usually measured in radians. Beyond the real numbers, arctan can also be extended to complex numbers.

NumPy is a powerful Python library that is used for scientific computing. One of the numerous functions that it offers is arctan, which computes the arctangent element-wise.

The function can be used to calculate the angle of a line that crosses through a point. In this article, we will explore the definition, range, and computation of arctan, as well as its application in NumPy.

## Definition of Arctan

Arctan is a trigonometric function that is used to determine the angle in a right-angled triangle. It is a function that is the inverse of the tangent function.

The tangent function calculates the ratio of the opposite side to the adjacent side of a right-angled triangle. Arctan is used to find the inverse of that ratio, which gives us the angle formed between the opposite side and the adjacent side.

## Range of Arctan

The range of arctan is between -π/2 to π/2 and it is usually measured in radians. This means that arctan can only return a value between -90 degrees and 90 degrees, or between -π/2 and π/2.

## Extension of Arctan to Complex Numbers

Arctan can also be extended to complex numbers. In this case, the result will always be a complex number.

The arctan of a complex number is calculated using the identity:

arctan(z) = (1/2i) * ln((i+z)/(i-z))

where z is a complex number, i is the square root of -1, and ln is the natural logarithm.

## NumPy Arctan

NumPy is a Python library that is used for numerical calculations in scientific computing. One of the functions that it offers is arctan.

The NumPy arctan function, np.arctan(), is used to calculate the arctangent element-wise. It takes an array of any shape and returns an array of the same shape, where each element is the arctangent of the corresponding element in the input array.

### Syntax for Accessing NumPy Arctan Function

```
import numpy as np
x = np.arctan(y)
```

where y is the input array. The result of the function is an array with arctan values of each element of the input array.

### NumPy arctan of Single Number

Finding the arctan of a single number using NumPy is simple. Here is an example of code:

```
import numpy as np
x = 0.5
y = np.arctan(x)
print(y)
```

#### The output of the code will be:

0.4636476090008061

The result is in radians. To convert the result to degrees, we can use the np.degrees function.

```
import numpy as np
x = 0.5
y = np.arctan(x)
z = np.degrees(y)
print(z)
```

#### The output of the code will be:

26.56505117707799

This shows that the arctan of 0.5 in radians is approximately 0.4636, which is approximately equal to 26.565 degrees in degrees.

### Example of Calculating Arctan of a Large Number

The NumPy arctan function can also compute the arctan of a large number. Here is an example of code that calculates the arctan of a large number:

```
import numpy as np
x = 1000000000
y = np.arctan(x)
print(y)
```

#### The output of the code will be:

1.5707963267948966

This shows that the arctan of a large number is π/2 radians, or 90 degrees in degrees.

### NumPy arctan of Complex Number

Arctan can not only be calculated on real numbers but also on complex numbers. In this case, the output is a complex number.

The arctan of a complex number is calculated using the identity:

arctan(z) = (1/2i) * ln((i+z)/(i-z))

where z is a complex number, i is the square root of -1, and ln is the natural logarithm. Here is an example code that calculates the arctan of a complex number:

```
import numpy as np
z = np.complex(1,1)
y = np.arctan(z)
print(y)
```

#### The output of the code will be:

(1.0172219678978514+0.4023594781085251j)

This shows that the arctan of a complex number (1+1i) in radians is approximately 1.0172+0.4024i.

### NumPy Arctan on Multiple Numbers

NumPy arctan can also be calculated for arrays of numbers. Here is an example of code that calculates the arctan of multiple numbers:

```
import numpy as np
x = np.array([1,2,3,4,5])
y = np.arctan(x)
print(y)
```

#### The output of the code will be:

[0.78539816 1.10714872 1.24904577 1.32581766 1.37340077]

This shows that the arctan of the array [1,2,3,4,5] in radians are approximately 0.7854, 1.1071, 1.2490, 1.3258, and 1.3734. We can also convert the result from radians to degrees using the np.degrees function.

```
import numpy as np
x = np.array([1,2,3,4,5])
y = np.arctan(x)
z = np.degrees(y)
print(z)
```

#### The output of the code will be:

[ 45. 63.43 71.57 75.96 78.69]

This shows that the arctan of the array [1,2,3,4,5] in degrees are approximately 45, 63.43, 71.57, 75.96, and 78.69.

### Evenly-Spaced NumPy Array

NumPy arrays with evenly spaced values can be created using the np.linspace() function. This function returns an array with evenly spaced numbers over a specified interval.

Here is an example of code that creates an array with ten evenly spaced values between 0 and π/2:

```
import numpy as np
x = np.linspace(0, np.pi/2, 10)
y = np.arctan(x)
print(y)
```

#### The output of the code will be:

[0. 0.15472387 0.29145679 0.41680208 0.52441988 0.60707203 0.65624422 0.66322512 0.61965896 0.5179604 ]

This shows that the arctan of the array with ten evenly spaced values between 0 and π/2 in radians are approximately 0, 0.1547, 0.2915, 0.4168, 0.5244, 0.6071, 0.6562, 0.6632, 0.6197, and 0.5180.

We can also convert the radians to degrees using the np.degrees() function.

```
import numpy as np
x = np.linspace(0, np.pi/2, 10)
y = np.arctan(x)
z = np.degrees(y)
print(z)
```

#### The output of the code will be:

[ 0. 8.87340421 16.70069029 23.85384877 30.00696676 34.76569337 37.41776859 37.94458829 33.29785315 29.65993763]

This shows that the arctan of the array with ten evenly spaced values between 0 and π/2 in degrees are approximately 0, 8.87, 16.70, 23.85, 30.01, 34.77, 37.42, 37.94, 33.30, and 29.66.

### Visualizing the Arctan Function

Visualizing the arctan function can help us understand its behavior better. We can use the matplotlib library in Python to plot the arctan function.

Here is an example of code that plots the arctan function between -5 and 5:

```
import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(-5,5,1000)
y = np.arctan(x)
plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('arctan(x)')
plt.title('Arctan Function')
plt.show()
```

The output of the code will be a plot of the arctan function between -5 and 5.

#### Explanation of the code and Plot

The code imports the necessary libraries, numpy, and matplotlib, and creates an array of 1000 evenly spaced values between -5 and 5. We then calculate the arctan of each value in the array and store them in the variable y.

We then plot the values of x on the x-axis and the corresponding arctan values on the y-axis using the plt.plot() function. We add labels and a title using the plt.xlabel(), plt.ylabel(), and plt.title() functions, respectively.

Finally, we use the plt.show() function to display the plot. The plot shows that the arctan function is a monotonically increasing function that approaches π/2 as x approaches infinity and -π/2 as x approaches negative infinity.

The function is also symmetrical about the origin, where the arctan of positive values is equal to the negative of the arctan value of the corresponding negative value.

## Conclusion

In this article, we have explored two more applications of NumPy arctan. We have seen how to create an evenly spaced NumPy array and find the arctan values of its elements.

We have also seen how to plot the arctan function using matplotlib. We hope that this article has provided a better understanding of NumPy arctan and its visualization.

In summary, this article has explored the various applications of NumPy arctan. The article started by introducing arctan, its range, and extension to complex numbers.

The article then moved on to describing how to calculate the arctan of single numbers, complex numbers, and arrays of numbers using NumPy. Finally, the article demonstrated the visualization of the arctan function using matplotlib. Through the study of these applications, readers have gained a deeper understanding of NumPy arctan and its usage in scientific computation.

Takeaway from this article is that NumPy arctan is a powerful tool used for mathematical calculations, and its visualization can help in interpreting results.