Radians vs Degrees: Explaining to Cooper

[Coop]

Homework Statement

The voltage across a 60μF capacitor is described by the equation vC =(18 V) cos(200t), where t is in seconds.

What is the voltage across the capacitor at t =0.010 s?

Homework Equations

N/A

The Attempt at a Solution

When you use degrees the answer is 18V and when you use radians its -7.5 V (the correct answer). Can someone explain why you must use radians here? It's been a while since using these in math class so the simpler explanation the better if you wouldn't mind :) (maybe I am missing something obvious).

Thanks,
Cooper

 

[SteamKing]

For your formula, you should be using radians rather than degrees. The argument of the cosine is going to be a pure number without units, which suggests that radian measure is the one to use.

 

[Coop]

For your formula, you should be using radians rather than degrees. The argument of the cosine is going to be a pure number without units, which suggests that radian measure is the one to use.

Why would using degrees not give a unitless number?

 

[jaytech]

You should be using radians because you are dealing with AC. This is obvious because the voltage is given as a cosine wave and alternating current is a waveform (whereas DC is approximated by a "linear" voltage). The AC has an angular frequency of 200 radians because the general equation for a cosine wave is Acos(kx+ωt).

 

[SteamKing]

Why would using degrees not give a unitless number?

Technically, a degree of angular measure is not strictly dimensionless; it represents part of the arc of a circle, but it cannot be decomposed into basic units of mass, length, or time. The radian, by definition, is the ratio of the length of a circular arc to the radius of the arc, and as such, is dimensionless (L/L).

http://en.wikipedia.org/wiki/Radian

 

[Chestermiller]

I have a different perspective. There is nothing fundamentally wrong with using degrees per second rather than radians per second. However, in physics, it is customary to work in terms of radians per second. Maybe that's because, if you wanted to calculate the rate of change of V with respect to time, you would have a real problem if you were working in terms of degrees/sec.

Chet