# Equation showing relationship between degrees and radians

### Degrees and radians

Can you have square or cubic degrees? Radians to the rescue! Knowing they refer to distance traveled (they're not just a ratio!), we can interpret the equation. In Mathematics we differentiate these as follows: Radian is a measure of an angle, indicating the relationship between the arc length and the radius of a circle . The denominator is the bottom number of a fraction. So, degree measure and radian measure are related by the equations ° = 2 π radians. and. ° = π.

The Unselfish Choice Much of physics and life! Instead of wondering how far we tilted our heads, consider how far the other person moved.

Degrees measure angles by how far we tilted our heads. Radians measure angles by distance traveled.

## Convert Radians to Degrees

So we divide by radius to get a normalized angle: Moving 1 radian unit is a perfectly normal distance to travel. Strictly speaking, radians are just a number like 1. Now divide by the distance to the satellite and you get the orbital speed in radians per hour. Sine, that wonderful function, is defined in terms of radians as This formula only works when x is in radians!

Well, sine is fundamentally related to distance moved, not head-tilting. Ok, the wheels are going degrees per second. Now imagine a car with wheels of radius 2 meters also a monster. Wow -- the car was easier to figure out than the bus! No crazy formulas, no pi floating around — just multiply to convert rotational speed to linear speed.

All because radians speak in terms of the mover. The reverse is easy too. How fast are the wheels turning? Calculus is about many thingsand one is what happens when numbers get really big or really small. Choose a number of degrees xand put sin x into your calculator: When you make x small, like. Even stranger, what does it mean to multiply or divide by a degree?

Can you have square or cubic degrees?

### Converting radians to degrees and degrees to radians

Radians to the rescue! How can we use this relationship now to figure out what degrees is?

Well, this relationship, we could write it in different ways. We could divide both sides by degrees, and we could get pi radians over degrees is equal to one, which is just another way of saying that there are pi radians for every degrees, or you could say, pi over radians per degree.

The other option, you could divide both sides of this by pi radians. You could say, you would get on the left hand side you'd get one, and you would also get, on the right hand side, you would get degrees for every pi radians. Or you could interpret this as over pi degrees per radian.

How would we figure out, how would we do what they asked us? Let's convert degrees to radians.

Let me write the word out. Well, we wanna convert this to radians, so we really care about how many radians there are per degree, actually, let me do that in that color. We'll do that same green color. How many radians are there per degree? Well, we already know, there's pi radians for every degrees, or there are pi Let me do that yellow color. There are pi over radians per degree.

And so, if we multiply, and this all works out because you have degrees in the numerator, degrees in the denominator, these cancel out, and so you are left with times pi divided by radians. So what do we get? This becomes, let me just rewrite it. All of that overso this is equal to, and we get it in radians.

And so, if we simplify it, let's see, we can divide the numerator and the denominator both by, looks like, So if you divide the numerator by 30, you get five.

You divide the denominator by 30, you get six. Now let's do the same thing for negative 45 degrees. What do you get for negative 45 degrees if you were to convert that to radians? You have negative, and I'll do this one a little quicker. I'll write down the word. Times, times pi radians, pi radians for every degrees.