Trigonometric Functions in Python - sin, cos, tan etc

The math library in python has a plethora of trigonometric functions which are enough for performing various trigonometric calculations in just minimal lines of code. These functions can be used after importing the math module or by referencing the math library with the dot operator as follows:

math.function_name(parameter)

# or import the math module

import math

There are 2 helper methods that help convert values from radians to degrees and vice-versa so that you can check the value in any format you want.

degrees(value): This method converts the value passed to it from radians to degrees.

radians(value): This method converts the value passed to it from degrees to radians.

Quick Fact: All the trigonometric functions in Python assume that the input angle is in terms of radians.

Also, as we study mathematics, pi/2 90 degrees and pi/3 is 60 degrees, the math module in python provides a pi constant that represents pi which can be used with the trigonometric function.

Most Common Trigonometric Functions in Python

Time for an Example:

In the code example below, we have used the degrees() and radians() methods,

import math

print(math.degrees((math.pi/2)))
print(math.radians(60))

Output:

90.0
1.0471975511965976

Now let's see the various trigonometric functions in action, one by one.

sin(x) Function

This function returns the sine of the value which is passed (x here). The input x should be an angle mentioned in terms of radians (pi/2, pi/3/ pi/6, etc).

cos(x) Function

This function returns the cosine of the value passed (x here). The input x is an angle represented in radians.

tan(x) Function

This function returns the tangent of the value passed to it, i.e sine/cosine of an angle. The input here is an angle in terms of radians.

Code example for sin, cos, and tan:

Below we have a simple code example for the 3 trigonometric functions defined above,

import math

print(math.sin(math.pi/3)) #pi/3 radians is converted to 60 degrees
print(math.tan(math.pi/3))
print(math.cos(math.pi/6))

Output:

0.8660254037844386
1.7320508075688767
0.8660254037844387

asin(x) Function

This function returns the inverse of the sine, which is also known as the arc sine of a complex number. The input is in terms of radians and should be within the range -1 to 1. It returns a floating-point number as output.

acos(x) Function

This function returns the cosine inverse of the parameter x in radians. This is also known as the arc cosine of a complex number. The value of x should be within the range of -1 and 1. It returns a floating-point number as output.

atan(x) Function

This function returns the inverse of the tangent, in radians. It is known as the arc tangent of the complex number. The parameter's value lies within -1 and 1. It can also be represented as (math.sin(math.pi / 4)) / (math.cos(math.pi / 4)). It returns a floating-point number as output.

Code example for asin, acos, and atan with angles in radians:

import math

print(math.asin(1))
print(math.acos(0))
print(math.atan(1))

Output:

1.5707963267948966
1.5707963267948966
0.7853981633974483

The below example depicts how the asin, acos and atan functions can be used with complex numbers, which have the format x+iy.

import cmath

x = 1.0
y = 1.0
z = complex(x,y)

print ("The arc sine is: ",cmath.asin(z))
print ("The arc cosine is: ",cmath.acos(z))  
print ("The arc tangent is: ",cmath.atan(z))

Output:

The arc sine is : (0.6662394324925153+1.0612750619050357j)
The arc cosine is : (0.9045568943023814-1.0612750619050357j)
The arc tangent is : (1.0172219678978514+0.40235947810852507j)

sinh(x) Function

This method returns the hyperbolic sine of the angle of the complex number that is passed.

cosh(x) Function

This method returns the hyperbolic cosine of the angle of the complex number that is passed.

tanh(x) Function

This method returns the hyperbolic tangent of the angle of the complex number that is passed.

Code example for sinh, cosh, and tanh with complex numbers:

import cmath

x = 1.0
y = 1.0
z = complex(x,y)
print ("The hyperbolic sine is : ",cmath.sinh(z))
print ("The hyperbolic cosine is : ",cmath.cosh(z))  
print ("The hyperbolic tangent is : ",cmath.tanh(z))

Output:

The hyperbolic sine is : (0.6349639147847361+1.2984575814159773j)
The hyperbolic cosine is : (0.8337300251311491+0.9888977057628651j)
The hyperbolic tangent is : (1.0839233273386946+0.2717525853195118j)

asinh(x) Function

It returns the inverse of the hyperbolic sine of an angle/complex number.

acosh(x) Function

It returns the inverse of the hyperbolic cosine of an angle/complex number.

atanh(x) Function

It returns the inverse of the hyperbolic tangent of an angle/complex number.

Code example for asinh, acosh, and atanh methods with complex numbers:

import cmath

x = 1.0
y = 1.0
z = complex(x,y)

print ("The inverse hyperbolic sine is : ",cmath.asinh(z))
print ("The inverse hyperbolic cosine is : ",cmath.acosh(z))  
print ("The inverse hyperbolic tangent is : ",cmath.atanh(z))

Output:

The inverse hyperbolic sine is : (1.0612750619050357+0.6662394324925153j)
The inverse hyperbolic cosine is : (1.0612750619050357+0.9045568943023813j)
The inverse hyperbolic tangent is : (0.40235947810852507+1.0172219678978514j)

atan2(y, x) Function

This translates to atan2(y/x) wherein both y and x are numeric values. It returns a value that lies between –pi and pi. This represents the angle between the positive x-axis and the coordinates (x,y).

print(math.atan2(1.2,1.3))

Output:

0.7454194762741583

hypot(x,y) Function

This function returns the hypotenuse, i.e the Euclidean norm. This means it returns the distance between the origin and the points (x,y). This indicates the length of the vector in the 2-D space. The Euclidean norm is also known as the ‘magnitude’ of the vector. This is the numerical value of the vector (since vectors have magnitude and direction).

If more than 2 arguments are passed, it gracefully returns a TypeError.

print(math.hypot(2,2))

Output:

2.8284271247461903

NOTE: When these values are compared to the numeric values of the angles, there might be a few decimal place differences, which can be safely ignored. For example, tan(pi/3) converts to tan 60 and which is equivalent to sqrt(3). If the tan function is used, the output is 1.7320508075688767 whereas sqrt(3) gives 1.7320508075688772.

Is it this simple? What about exceptions and errors?

Well, you are absolutely right. There are two kinds of exceptions that could potentially occur due to the wrong usage of these methods:

TypeError: This kind of error occurs when a value that is not a number is passed as a parameter to one of the trigonometric methods.

ValueError: This error occurs when an invalid value/parameter is passed to the method.

An example demonstrating both exceptions:

import math

# It is being passed as a string rather than a number, 
# hence the error.
print(math.sin('2')) 

# Inverse of the sine angle can be found for parameters 
# which are in radians only and inverse of sine of an integer 
# is mathematically undefined, leading to the error
print(math.asin(5)) 

Output:

TypeError: must be real number, not str
ValueError: math domain error

Conclusion

We learned about how various trigonometric methods could be used by importing the math module in python. Remember to always pass the parameter in radians or convert the degrees into radiansand then pass it as a parameter.

Frequently Asked Questions (FAQs) :

1. Does Python have trigonometric functions?

Python has a wide range of Trigonometric functions like math.cos(), math.sine() that can be used to calculate the Trigonometry terms like these. Read the complete article to know how you can use Trigonometric functions in Python.

2. What are cos functions in Python?

math.cos() function in Python can give the cosine of a number.

3. How do you define tan in Python?

tan() function is used to to calculate the tangent of a number.

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