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Radians

What is a radian?

Graphically defining a radian.

Consider a circle of radius r as shown to the right. Now, mark off a portion of the circumference such that the selected distance around the circumference is equal to r. This portion of the circumference defines an angle, which we have labelled in the figure with the lower-case Greek letter phi (φ, pronounced "fee"). Because the portion of the circumference that is enclosed by this angle is equal in length to the radius of the circle, the angle itself is designated as one radian.

We already know that the entire circumference, C, of the circle is give by the expression C = 2πr, where π is the lower-case Greek letter pi, and represents a value of approximately 3.14159265. We also know that the entire circle encompasses a total of 360°. Therefore we can calculate the angle of one radian in terms of degrees as follows:

2 × π × φ  =  360°
 
φ  =  360°

2 × π
 
   =  360°

6.2831853
 
   =  57.29578°


Where Do We Use Radians?

In electronics, we commonly use radians in two ways:



Why Use Radians?

While it is certainly possible to make all of our calculations for electronic circuits in terms of hertz rather than radians per second, there is one extremely practical reason to stick with radians except at the very beginning and the very end of your procedure, and maybe avoid hertz altogether: use of radians simplifies the mathematical calculations.

This is because the factor 2π appears very often in calculations involving hertz, but is eliminated from those same calculations involving radians. By avoiding that factor throughout a circuit analysis or design process, we can reduce the chances of mathematical errors. Also, since π is a transcendental number — a number with no final resolution — using its approximate value many times in your calculations can lead to accumulating roundoff errors and a gradual drift away from accurate numbers.

By performing our calculations in radians rather than cycles or degrees, and then converting back and forth only at the very start and end of the procedure, we bypass these issues and maintain greater precision in our computations, and a more accurate final result.


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