Identifying reference angles of positive angles in standard position

0:00- [Instructor] So let's talk a little bit about reference angles. So these are angles that are formed between, you could view it as the terminal ray of an angle and the x-axis that is positive and acute, and we usually denote it with a theta prime. Now you might say, "Why do we even need this thing called a reference angle?"

0:24And you'll see, as we expand our definition of trigonometric functions, this will be very useful for being able to evaluate different trigonometric functions to really extend what you already know about them. So let's just conceptually think about the reference angles in each of these situations.

0:39So here, I have an angle that's in the first quadrant. Already, it's a positive and acute angle. And so it's a positive acute angle formed with the x-axis. So here's a situation where this is also going to be the reference angle. So theta is equal to theta prime.

0:56So this was, for example, a 45-degree angle. Well, the reference angle is also 45. If we were to say this in terms of radians, if this were pi/4 radians, pi over 4 radians, then this will also be the reference angle. Now it gets a little bit more interesting when we go into the other quadrants.

1:16This here is in the second quadrant. So what would be the positive acute angle formed with the x-axis? Well, it would be this right over here. That would be the reference angle. Now how would we compute it? Let me make that clear. That's a prime right over there.

1:32How would we compute that? Let's say if theta, in this situation. theta is equal to 130 degrees, well, in this case, the reference angle is taking us all the way to 180. So it's gonna be 180 degrees minus that. So I write theta prime, in this situation, is 180 degrees - 130 degrees, which is equal to 50 degrees.

1:59This is a positive acute angle between we could say that terminal ray and the x-axis. And as I said, this will be useful in the future when you're trying to evaluate trig functions or where that extension of your current definition of trig functions come from.

2:15Let's go into the third quadrant, and we could have done this with radians as well. Actually, maybe I'll do this example in radians. So here we have, let me, so here we have an angle that gets us into the third quadrant right over there. And what would be the reference angle?

2:35Well, the reference angle, once again, it's an acute positive angle formed with the x-axis. So acute positive angle formed with the x-axis would be this right over there. Now let me construct a theta in radians that matches. So we know that this would be, if we're pointed this way, we're at zero radians.

2:53This is pi over 2. This is pi. This is 3pi over 2. And so e could view this as 2pi over 2. So we could view this as maybe 2 1/2pi over 2, which is the same thing as 5pi over 4. So if theta, in this situation, is equal to 5pi over 4, 5pi over 4, what would this reference angle be?

3:22Well, it would be this minus either 180 degrees if we were thinking in degrees or minus pi. So here our reference angle, theta prime, is equal to 5pi over 4 - pi, which is the same thing as 4pi over 4. And so this will be equal to pi over 4. That is our reference angle, positive acute angle with the x-axis.

3:51Now last but not least, we are in the fourth quadrant. What would our reference angle be? Well, it would be this right over here, positive acute angle with the x-axis. And let me give you an example. Let's say that theta here is 300, and let me say 340 degrees.

4:12I'm switching back to degrees now. Well, theta prime, it would be what takes you all the way to 360. So theta prime is essentially going to be 360 degrees minus 340 degrees, which is equal to 20 degrees. So hopefully, you have a sense of how to compute reference angles, what they are conceptually, and you'll see in the future, this will make it much simpler to evaluate trig functions conceptually.