Identifying reference angles for negative angles in standard position
About this video
The video finds reference angles for negative angles, which rotate clockwise from the positive x-axis. Negative 700 degrees goes almost two full turns and stops 20 degrees short of 720, so its reference angle is 20 degrees; negative 13pi/6 goes one full turn (negative 2pi) plus another pi/6, so its reference angle is pi/6.
You'll be able to
- sketch a negative angle by rotating clockwise from the positive x-axis.
- find the reference angle of a negative angle in degrees or radians.
Key words
- negative angle
- An angle measured clockwise from the positive x-axis.
- reference angle
- The positive acute angle between an angle's terminal ray and the x-axis.
- clockwise
- The direction the hands of a clock turn.
- radian
- A unit for measuring angles; a full turn is 2pi radians, or 360 degrees.
Transcript
- [Instructor] What we're gonna do in this video is think a little bit more about reference angles. and in particular, we're gonna try to figure out the reference angles for negative angles. So let's do a negative angle in degrees, so let's say we're thinking about negative 700 degrees, so what would that look like?
Well, we could start at the positive X axis, and if we're talking about a negative angle, we'd be going in the clockwise direction, so if we go around once, that is 360 degrees, and then if we were to go around a whole other time, it would be 720, so we're going to be 20 degrees short of that.
So what I just drew right over there, that is negative 700 degrees, and the angle would look like this. Maybe I'll do it in this purple color. It would look something like this. So one way to think about it is, you would be starting at the X axis, and you'd be going around almost twice to get back over there.
Now, as we've talked about before, reference angles, which are going to be very useful as we go further into trigonometry, are positive acute angles, what we are talking about with the X axis, so for example, in this situation right over here, that would be this angle that goes from the X axis to that ray right over there, and as we said, this is 20 degrees.
It's 720 minus that 700 degrees, so that is our reference angle, our positive angle. Now, let's do it in radians. So let's say we're dealing with negative 13 over, let's say, negative 13 pi over six radians, radians. What does that look like? Well, once again, we can start at the positive X axis.
This would involve, let's see, 12 pi over six would be going twice around, so let's see. If we go, this would be negative two pi over, this would be negative two pi, which is the same thing. Well, actually, nevermind. That would be, so what I just drew so far is negative two pi, which is 12 pi over six, so now, we are going to go another pi over six.
So another way to think about it, we'd go another pi over six, and it might look something like this. So the angle, so how did I do that? This angle up here, negative 13 pi over six, that's the same thing as negative 12 pi over six minus another pi over six. This is negative two pi, negative two pi minus pi over six, so negative two pi, you'd go one time around, it's equivalent of negative 360 degrees, and then you go another negative pi over six.
Now, what is the reference angle going to be? Well, remember, it's a positive angle with the X axis, so we are going to be talking about, it is going to be, we are going to be talking about this angle right over here, and that is going to be pi over six radians.
It's, again, positive angle with the X axis, and we're done.
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