A complete rotation in a circle causes the rotator to face in the same direction after rotation as before it. The angle rotated through is two 𝜋 radians which equals 360 degrees. How many radians is 1.0 degrees equal to?
The statement tells us that if we have a circle and we sweep through a complete rotation through the circle, then that angle is equal to two 𝜋 radians, which is also equal to 360 degrees. Since those two values are equal, that means that if we divide two 𝜋 radians by 360 degrees, that this fraction is equal to one. With units of radians per degree, we want to solve for how many radians 1.0 degrees is equal to. So we’ll want to convert this value to another value in a different unit. Notice that if we multiply a measure in degrees by a conversion factor in radians per degree, then we get a result in radians.
So let’s multiply two 𝜋 radians per 360 degrees by 1.0 degrees. Notice that the units of degrees cancel out and we’re left with the units of radians, just as we want. Two 𝜋 radians times 1.0 divided by 360 equals 0.017 radians. That’s how many radians are in 1.0 degree. And this conversion factor we developed works for any conversion from degrees to radians.