# Radians vs Degrees: Explaining to Cooper

• Coop
In summary, when dealing with AC and using the equation vC =(18 V) cos(200t), it is important to use radians rather than degrees to get a unitless number for the argument of the cosine. This is because AC is a waveform and the AC in this equation has an angular frequency of 200 radians. While degrees can be used for angular measure, it is customary in physics to work in terms of radians per second for consistency and ease of calculation.
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?

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

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.

SteamKing said:
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?

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).

Coop said:
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).

1 person
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

## What is the difference between radians and degrees?

Radians and degrees are two units used to measure angles. Degrees are commonly used in everyday life, while radians are more commonly used in mathematics and science. One radian is equal to 57.3 degrees.

## How do you convert radians to degrees?

To convert radians to degrees, simply multiply the number of radians by 180 and divide by pi. For example, if you have 2 radians, the conversion would be (2 * 180) / pi = 114.59 degrees.

## Why do we use radians in mathematics and science?

Radians are preferred in mathematics and science because they simplify many mathematical formulas and calculations. This is because radians are based on the radius of a circle, making it easier to work with trigonometric functions.

## Which is a better unit for measuring angles, radians or degrees?

Neither unit is inherently better than the other. It depends on the context and what is being measured. For example, degrees may be more useful for measuring everyday angles, while radians may be more useful for complex mathematical calculations.

## Are radians and degrees interchangeable?

Yes, radians and degrees are interchangeable. However, it is important to note that they are two different units of measurement and should be used appropriately depending on the situation. It is possible to convert between the two units, but they are not equal to each other.

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