Remember that the fastest way to convert Hertz to radians per second is to simply multiply the number of Hertz by 6.28. A pattern with a frequency of 2 Hertz can also be expressed as a pattern with a frequency of 12.56 radians per second. When frequency is expressed in radians per second, it is often referred to as angular frequency.
A repeating pattern is said to rotate through 360 degrees--a circle--over each repetition of the pattern. Think of a radian as a slice of pie, a piece of that circle. The angle formed by the end of the slice intercepts a section of the circumference, or crust. In a radian, the radius equals the length of the arc of the circumference. This is true when the angle is 57.3 degrees. Since there are 360 degrees in a circle, there are 6.28 radians in a circle. (360/57.3 = 6.28).
Convert Hertz to cycles per second by multiplying the frequency in Hertz by one. A waveform with a frequency of 1,000 Hertz has a frequency of 1,000 cycles per second.
Convert cycles per second to radians per second. Multiply the frequency in cycles per second by 6.28 radians per cycle. Note that when you multiply these two ratios, the cycle units cancel out, leaving you with radians per second. A waveform with a frequency of 1 cycle per second will have an angular frequency of 6.28 radians per second (6.28 x 1 = 6.28). A waveform with a frequency of 2,000 cycles per second will have an angular frequency of 12,560 radians per second (6.28 x 2,000 - 12,560).
Understand that angular frequency is indirectly based on the number of degrees in a circle. If a wheel makes one complete revolution in one second it is said to have an angular frequency of 360 degrees per second. A circle that rotates at one cycle per second is also said to rotate at 6.28 radians per second.