Is there a rule for when to use 2Pi/b = period or 360/period?

Question

When writing equations for sine or cosine application problems, sometimes the instructor leaves the solution in radians and sometimes they divide 360 by the period. Is there a rule for when to use one method over the other? Do they provide the same solutions?

Answer 1

There may be a little confusion here. To begin with both degrees and radians are a measure of angle. So 2Pi = 360 degrees, which is one full rotation starting from a vector pointing in the + direction along the x-axis and then going ccw around once through all four quadrants.

The confusing part is the use of T and b. I presume that b = T and that each is a symbol for the period in seconds. If this is the case, then 2Pi/T = 360/b and both are the same thing, namely the angular frequency, one measured in radians per second and the other in degrees per second. Note that the usual symbol for angular frequency is the Greek letter omega.

So, as to which one to use that's arbitrary. Usually the problem statement will specify what is desired. And of course, when using your calculator, you'll have to be sure that when you want to find the sine of an angle you are not entering degrees, when the calculator is in radian mode.

Answer 2

I'm not aware of any applicable rules. What you are discussing seems more like conventions, which could be thought of as arbitrary definitions to avoid confusion. The most common example is probably "conventional current" which flows from + to -.

One thing IS a rule: 180 degrees = π radians

Sinusoidal functions are often written in the form y = A*sin(ωt + θ), where A is the amplitude (0 to peak), ω is angular frequency (radians/sec), t is time, and θ is phase. ω = 2*π*f, where f is frequency. Any way you do it, the argument for the sine (or cosine) function needs to be an angle.

For angles, it does not matter if you use radians or degrees, as long as you are consistent within a problem.