Python includes two functions in the math package; radians converts degrees to radians, and degrees converts radians to degrees.

To match the output of your calculator you need:

>>> math.cos(math.radians(1))

0.9998476951563913

Note that all of the trig functions convert between an angle and the ratio of two sides of a triangle. cos, sin, and tan take an angle in radians as input and return the ratio; acos, asin, and atan take a ratio as input and return an angle in radians. You only convert the angles, never the ratios.

Python convert radians to degrees or degrees to radians:

What are Radians and what problem does it solve?:

Radians and degrees are two separate units of measure that help people express and communicate precise changes in direction. Wikipedia has some great intuition with their infographics on how one Radian is defined relative to degrees:

https://en.wikipedia.org/wiki/Radian

Python examples using libraries calculating degrees from radians:

>>> import math

>>> math.degrees(0) #0 radians == 0 degrees

0.0

>>> math.degrees(math.pi/2) #pi/2 radians is 90 degrees

90.0

>>> math.degrees(math.pi) #pi radians is 180 degrees

180.0

>>> math.degrees(math.pi+(math.pi/2)) #pi+pi/2 radians is 270 degrees

270.0

>>> math.degrees(math.pi+math.pi) #2*pi radians is 360 degrees

360.0

Python examples using libraries calculating radians from degrees:

>>> import math

>>> math.radians(0) #0 degrees == 0 radians

0.0

>>> math.radians(90) #90 degrees is pi/2 radians

1.5707963267948966

>>> math.radians(180) #180 degrees is pi radians

3.141592653589793

>>> math.radians(270) #270 degrees is pi+(pi/2) radians

4.71238898038469

>>> math.radians(360) #360 degrees is 2*pi radians

6.283185307179586

Source: https://docs.python.org/3/library/math.html#angular-conversion

The mathematical notation:

You can do degree/radian conversion without python libraries:

If you roll your own degree/radian converter, you have to write your own code to handle edge cases.

Mistakes here are easy to make, and will hurt just like it hurt the developers of the 1999 mars orbiter, who sunk $125m dollars crashing it into Mars because of non intuitive edge case here.

>>> 0 * 180.0 / math.pi #0 radians is 0 degrees

0.0

>>> (math.pi/2) * 180.0 / math.pi #pi/2 radians is 90 degrees

90.0

>>> (math.pi) * 180.0 / math.pi #pi radians is 180 degrees

180.0

>>> (math.pi+(math.pi/2)) * 180.0 / math.pi #pi+(pi/2) radians is 270 degrees

270.0

>>> (2 * math.pi) * 180.0 / math.pi #2*pi radians is 360 degrees

360.0

Degrees to radians:

>>> 0 * math.pi / 180.0 #0 degrees in radians

0.0

>>> 90 * math.pi / 180.0 #90 degrees in radians

1.5707963267948966

>>> 180 * math.pi / 180.0 #180 degrees in radians

3.141592653589793

>>> 270 * math.pi / 180.0 #270 degrees in radians

4.71238898038469

>>> 360 * math.pi / 180.0 #360 degrees in radians

6.283185307179586

Expressing multiple rotations with degrees and radians

Single rotation valid radian values are between 0 and 2*pi. Single rotation degree values are between 0 and 360. However if you want to express multiple rotations, valid radian and degree values are between 0 and infinity.

>>> import math

>>> math.radians(360) #one complete rotation

6.283185307179586

>>> math.radians(360+360) #two rotations

12.566370614359172

>>> math.degrees(12.566370614359172) #math.degrees and math.radians preserve the

720.0 #number of rotations

Collapsing multiple rotations:

You can collapse multiple degree/radian rotations into a single rotation by modding against the value of one rotation. For degrees you mod by 360, for radians you modulus by 2*pi.

>>> import math

>>> math.radians(720+90) #2 whole rotations plus 90 is 14.14 radians

14.137166941154069

>>> math.radians((720+90)%360) #14.1 radians brings you to

1.5707963267948966 #the end point as 1.57 radians.

>>> math.degrees((2*math.pi)+(math.pi/2)) #one rotation plus a quarter

450.0 #rotation is 450 degrees.

>>> math.degrees(((2*math.pi)+(math.pi/2))%(2*math.pi)) #one rotation plus a quarter

90.0 #rotation brings you to 90.

Fundamental education on Radians and Degrees

5 minute refresher using Trigonometry and expression of rotation to convert radians to degrees and back: https://youtu.be/ovLbCvq7FNA?t=31

Khan academy refresher on trigonometry, unit circle, angular mathematics to use sin,cos,tan to describe rotation and changes in rotation. https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:trig/x2ec2f6f830c9fb89:unit-circle/v/unit-circle-definition-of-trig-functions-1