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Radians

Engineers tend to prefer using degrees (o) to radians (rad), if for no other reason, they are more comfortable with degrees: 360 is easy to remember, and easy to divide, being equal to 23⋅32⋅5.

The following are three arguments why you should use radians instead of degrees.

Derivatives

If x is measured in radians, then the derivative of sin(x) and cos(x) are cos(x) and -sin(x), respectively.

If x is measured in degrees, then the derivatives of sin(x) and cos(x) are 180/π cos(x) and -180/π sin(x), respectively.

Approximations

If x is measured in radians, then:

  • sin(x) ≈ x
  • cos(x) ≈ 1 - ½ x2
  • tan(x) ≈ x

If x is measured in radians, the formulae require appropriate constant multipliers.

Complex Exponential Function

The expanded version of the exponential function when using a complex exponent is of the form:

where the arguments of sin and cos are measured in radians.

Milliradians (mils)

As for the comment that radians are too difficult to learn:

The Canadian army uses radians, or a good approximation thereunto. The mil is defined such that there are 6400 mils to a circle, and therefore one mil is a good approximation to one milliradian (mrad) (with a relative error of less than 2%). Soldiers actually use the approximation tan(x) ≈ x when they use the fact that an error of n mils when calculating a bearing results in an error of n m at a distance of 1 km (1 km · tan(0.001 n) ≈ 0.001n km = n m). If infantry soldiers can learn to use (and love) radians (even if they don't know it) then an engineer can, too.