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### Course: Trigonometry>Unit 2

Sal explains the definition and motivation for radians and the relationship between radians and degrees. Created by Sal Khan.

## Want to join the conversation?

• Under what circumstances is it preferable to use radians instead of degrees? I could understand it being in architecture, but is it also used in other sciences?
• This is a question that bothered me for quite a few years while taking college physics and astronomy and I wish I would have learned this sooner. Andrew is correct, that radians are used heavily in physics and engineering. Here is an important circumstance that requires radians instead of degrees:

Anytime you're plugging an angle into some equation. For example, 4 + 45degrees makes absolutely no sense because the units do not match. Notice that 45degrees = (pi/4). Now, 4 + (pi/4) makes complete sense because (pi/4) is an actual number, it's a distance. Radians are basically just a unit of circular distance. A basic rule of thumb I found is that degrees are useful as long as they
(2) stay inside a trig operator, like Sin, Cos, Tan, ArcSin, etc.......
• Throughout the video, why does Sal keep saying "Radiuseseseseses..."? Why not "Radii"?
• I think it's because Sal has a sense of fun, and was playing with the word "radiuses". Foolishly, perhaps, thinking that others might share his sense of humour. A mistake it seems, since this question comes up so often.

• The radius is the distance from the center of a circle to its perimeter.
A radian is an angle whose corresponding arc in a circle is equal to the radius of the circle.
• How do I write pi symbol in the excercises answers?
• When you click where to type in your answer it comes up with little buttons underneath that you can click on and one of them is the pi symbol
• Why don't we just express a full rotation as 1 tau?
• Historical convention. π has been used historically much more than 𝜏, so there are few people in the world that actually understand if you start using 𝜏 instead of π.

Other than that, if you are certain whoever is reviewing your work will understand, you can use `𝜏` to represent a full rotation.
• Is there a notation for radian like there is that tiny circle for degrees?
• No, nor is there any need. Remember that an angle in radians is just the ratio of that arclength to the radius, so the units cancel out.

To avoid confusion, you can write "radians" if you like, but it is not needed nor customary. If no angle unit is mentioned, the units are always taken to be in radians. In higher math, radians are used almost exclusively, so it is exceedingly rare to even deal with degrees one you get into calculus.
• Why does Sal say radiuseses?
• Why is the degree notation a small circle??
• The degree symbol was first used in the 1500s, and it seems to have originated as a superscript zero. Sort of like the ' and " used for for feet and inches, originated as superscript roman numerals. :)
• I am extremely confused on what radians are. Can anyone state what their importance is and what they are?
• Radians are used to measure angles. You might be more used to measuring angles with degrees, in which case it should help to think of radians as a different sized unit to measure the same thing. A 360 degree angle is the same as a 2pi radian angle. Radians start being used in geometry and trig as you start using the unit circle. I think marking them in the unit circle is a good way to visualize how they work, and how they can be a good unit to work with in that context.
• i dont understand any of this. can someone simplify this or just restate what they said to where i can understand it with ease?
• Degrees are one way to measure angles. There are 360 degrees in a full circle. So, a degree is defined as 1/360 of a circle.

Radians are an alternative way to measure angles. There are 2pi radians in a full circle. A radian is defined as the amount of radiuses that fit in an arc. If an angle is 3 radians, that means 3 radiuses would fit in the arc of the circle created by that angle. There are 2pi radians in a circle because 2pi * radius = circumference (perimeter all the way around a circle).

Since they are both equal to a full circle, 360 deg = 2pi rad. Or 180 deg = pi rad. We can multiply degrees by pi/180 to get radians. We can multiply radians by 180/pi to get degrees.