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## Get ready for AP® Calculus

# Radian angles & quadrants

Sal determines the quadrant at which a ray falls after a rotation by a certain measure of radians.

## Want to join the conversation?

- I'm really confused

I thought 2pi=360° so pi is 180° and so if you move 3pi you will do 360 +180 degrees respectively or one rotation and a half but Sal puts it in the second quadrant

Can someone explain what I did wrong(17 votes)- Notice that there is a difference between 3 and 3pi. 3pi would indeed take you past 360 degrees, but 3 is a little less than just 1pi--thus putting it in quadrant 2.(33 votes)

- At3:10he writes the value of pi = 3.14, but Sal uses it as a reference for radians, isn't the 3.14 value in degrees? I mean he says that 3 radians is close to 3.14. So basically I'm asking what measure are we using in normal arithmetic, degrees right? If so, how come he compared 3 radians to 3.14 ,what are presumebly, degrees? Please tolerate any ignorance you come across in my question, I'm simply trying to understand the concepts.(2 votes)
- Pi is simply a mathematical constant. It does not have any default units attached to it. The number 5 is not in degrees or meters, it is just the number 5. The same with pi.(56 votes)

- At around2:35he says "pi/2 here would be 3.5 pi over 7". I have no clue what he means by that.(8 votes)
- He's trying to figure out if 2pi/7 is less than or greater than pi/2. By multiplying the numerator and denominator of pi/2 by 3.5, he gets 3.5pi/7. 3.5 is greater than 2, so 3.5pi/7 (or pi/2) is greater than 2pi/7, which means 2pi/7 would not pass pi/2 which puts it in the first quadrant.(8 votes)

- 1:18, why is 90 degrees equal to pi/2 radians?(2 votes)
`π radians = 180°`

`π/2 rad = 180°/2`

If you don't understand why radians are measured using pi, check out this animation:

http://en.wikipedia.org/wiki/User:LucasVB/Gallery#/media/File:Circle_radians.gif

Check out this one too for my preferred method of measuring radians, tau (τ):

http://en.wikipedia.org/wiki/User:LucasVB/Gallery#/media/File:Circle_radians_tau.gif(18 votes)

- Would it be quicker to just to convert from radian to degrees?(2 votes)
- I think solving the fraction would be a faster approach. Eg : 3pi/7

3/7= 0.4 approx. which is less than 0.5pi(=1st quadrant mark)

Therefore it lies in the first quadrant. :D(6 votes)

- what quadrant is pi radians in?(3 votes)
- If we rotate a ray π radians counter-clockwise, we'll get the ray lying on the x-axis.(5 votes)

- Sal said that pi radians= 180 degrees then can i say that 3.14 radians = 180 degrees?Help would be appreciated.

thank you(1 vote)- Pretty much, yes.... if you want to be "nit-picky" about it, 3.14 radians = 179.90875 degrees. In general, it's better to use pi instead of a rounded approximation like 3.14 or 22/7 or something like that.

By the way, 22/7 radians = 180.07245 degrees.(9 votes)

- At3:41shouldn't the terminal side rest on the on the line since 1 radian is a 180 degrees (Straight Line) and 2 radians (Full rotation). And if it has 3 radians shouldnt it be a full rotation and a straight line?(1 vote)
- Your conversions are incorrect.

1 radian is about 57.3°.

2 radians is about 114.6°

3 radians is about 171.9°

It is π radians that equals 180°

2π radians = 360°(9 votes)

- Going from this, how do we determine the measure of a coterminal angle in radian measure?(2 votes)
- angle+/- 2pi(k) or angle +/- 360(k) will give you the coterminal angles where k is an integer(3 votes)

- How do you know that 2pi/7 is less than pi/2?(2 votes)
- Let's say pi is 3 for right now, as an estimate. 2pi/7 would be 6/7, which is less than one. pi/2 would be 3/2, which is greater than one. If 2pi/7 < 1 < pi/2, than 2pi/7 is less than pi/2.

Since pi is actually 3.1415...... on and on forever, this is a pretty good estimate.

Hope this helps!(3 votes)

## Video transcript

- [Voiceover] What I
want to do in this video is get some practice, or
become familiar with what different angle measures in
radians actually represent. And to get our familiarity, we're gonna start with a ray
that starts at the origin, and moves along, and... Not moves, and points
along the positive X axis. We're gonna start with this magenta ray, and we're gonna rotate
it around the origin counterclockwise by
different angle measures. And think about what
quadrant do we fall into if we start with this and we
were to rotate counterclockwise by three pi over five radians? And then, if we start with this, and we were to rotate counterclockwise by two pi over seven radians? Or, if we were to start with this, and then rotate counterclockwise
by three radians? We encourage you to pause
the video and think about, starting with this, if we were to rotate
counterclockwise by each of these, what quadrant are we going to end up in? Assume you've paused the video, and you've tried it out on your own, so let's try this first
one, three pi over five. Three pi over five, so
we're gonna start rotating. If we go straight up, if we rotate it, essentially, if you want to think in degrees, if you rotate it
counterclockwise 90 degrees, that is going to get us to pi over two. That would have been a counterclockwise rotation
of pi over two radians. Now is three pi over five
greater or less than that? Well, three pi over five, three pi over five is greater than, or I guess another way I can say it is, three pi over six is less
than three pi over five. You make the denominator smaller, making the fraction larger. Three pi over six is the
same thing as pi over two. So, let me write it this way. Pi over two is less
than three pi over five. It's definitely past this. We're gonna go past this. Does that get us all the way over here? If we were to go, essentially, be pointed in the opposite direction. Instead of being pointed to
the right, making a full, I guess you could say 180 degree
counterclockwise rotation, that would be pi radians. That would be pi radians. But this thing is less than pi. Pi would be five pi over five. This is less than pi radians. We are going to sit, we are going to sit someplace, someplace, and I'm just estimating it. We are gonna sit someplace like that. And so we are going to sit
in the second quadrant. Let's think about two pi seven. Two pi over seven, do we
even get past pi over two? Pi over two here would
be 3.5 pi over seven. We don't even get to pi over two. We're gonna end up, we're gonna end up someplace, someplace over here. This thing is, it's greater than zero, so we're gonna definitely
start moving counterclockwise, but we're not even gonna get to... This thing is less than pi over two. This is gonna throw us
in the first quadrant. What about three radians? One way to think about it is, three is a little bit less than pi. Right? Three is less than pi but it's greater than pi over two. How do we know that? Well, pi is approximately 3.14159 and it just keeps going on and on forever. So, three is definitely closer to that than it is to half of that. It's going to be between
pi over two, and pi. It's gonna be, if we start
with this magenta ray, we rotate counterclockwise
by three radians, we are gonna get... Actually, it's probably gonna be, it's gonna look something, it's gonna be something like this. But for the sake of this exercise, we have gotten ourselves, once again, into the second quadrant.