Radians

 

We can measure Angles in Radians.

1 Radian is about 57.2958 degrees.

Does 57.2958... degrees seem a strange value?
Maybe degrees are strange, as the Radian is a pure measure based on the Radius of the circle:

Radian: the angle made by taking the radius
and wrapping it along the edge of a circle:

So, a Radian "cuts out" a length of a circle's circumference equal to the radius.

Radians and Degrees

So:

  • There are π radians in a half circle
  • And also 180° in a half circle

So π radians = 180°

So 1 radian = 180°/π = 57.2958° (approximately)

 

Degrees Radians
(exact)
Radians
(approx)
30° π/6 0.524
45° π/4 0.785
60° π/3 1.047
90° π/2 1.571
180° π 3.142
270° 3π/2 4.712
360° 2π 6.283

 

Example: How Many Radians in a Full Circle?

Imagine you cut up pieces of string exactly the length from the center of a circle to its edge ...

... how many pieces would you need to go around the edge of the circle?

 

Answer: 2π (or about 6.283 pieces of string).

Radians Preferred by Mathematicians

Because the radian is based on the pure idea of "the radius being laid along the circumference", it often gives simple and natural results when used in mathematics.

For example, look at the sine function for very small values:

x (radians) 1 0.1 0.01 0.001
sin(x) 0.8414710 0.0998334 0.0099998 0.0009999998

For very small values. "x" and "sin(x)" are almost the same
(as long as "x" is in Radians!)

There will be other examples like that as you learn more about mathematics.

Conclusion

So, degrees are easier to use in everyday work, but radians are much better for mathematics.