Radians Versus Degrees
This page compares and contrasts the two systems of measuring angles in math: radians and degrees, and explains why radians is the preferred unit of measure for angles. This page assumes a reasonable level of familiarity with the degrees system and an introductory level of experience with radians.
There are two systems for measuring degrees in mathematic: degrees and radians. The degree system divides a full roation into and often introduces the concept of an angle to students. The degree system is rooted in the first origins of trigonometry and is a useful system for measuring angles. However, as time has gone by and math progressed, the radian system has replaced degrees as the preferred system.
The short answer to why radians are preferred is that the radian system leads to more succinct and elegant formulas throughout mathematics^{[1]}. For example, the derivative of the sine function is only true when the angle is expressed in radians ^{[1]}.
The longer answer explains why the “dimensionless” or “radiusinvariant” property of radians leads to these more succinct and elegant formulas.
The concept of angles and circles are intimately related. Consider the angle formed by two rays as illustrated in figure 3 whose vertex is at the center of two cocentric circles of radii , and arclengths , . Observe that the angle can be defined in terms of the first circle and the second circle.
This observation suggests a natural definition for radians. In both cases, the arclength is proportional to the radius which implies that ratio of arclength to radius is the equal. This is stated more formally in math notation as:
This relationship gives us the definition of an angle measured in radians: A radian angle is equivalent to the ratio of arclength to radius. This definition of radians is expressed in the notation below.
As a result of this definition, a full rotation or “one turn” in radians is approximately equal to as shown in figure 4. This special number is called the circle constant and appears in many areas of math, statistics and physics.
The circle constant (tau) makes measuring angles in radians relatively straightforward and is where radians show up in numerous applications. A good rule of thumb is that if appears in formula, there usually is some notion of the circle present. The numeric value of can be calculated by dividing the circumference of any circle by its radius as shown below.
The advantages of the radian system become apparent when definining complex formulas such as the area of the circle, definition of the trigonometric function sine and cosine, normal distribution and Euler’s formula.
This section is still in development, the derivations are hopefully coming soon. Thanks for your patience!
TODO: derive area of circle formula
TODO: derive definition of sine and cosine
TODO: derive normal distribution formula
TODO: euler’s formula
Links
Radians are a unit that measure angle using the radius of a circle. One radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle.
Degrees is a unit of measure for angles. A full rotation is equal to 360 degrees. In the cartesian coordinate system, degrees are measured starting from the rightmost edge of the circle.
The circle constant τ (tau) is a geometric constant approximately equal to 6.283. The numeric value is defined as the length of any circle's circumference divided by the length of its radius.
References

No, really, pi is wrong: The Tau ManifestoMichael Hartl