Reference Angles - Video Tutorials & Practice Problems

On a tight schedule?

Get a 10 bullets summary of the topic

1

concept

Reference Angles on the Unit Circle

Video duration:

5m

Play a video:

Hey, everyone, we now know all of our trig values of our three common angles in quadrant one, but the unit circle has three other quadrants, all of which will need to know trick values in. Now, this can seem really overwhelming at first because there are a ton more angles here that we haven't even looked at yet. But you don't have to worry because for all of these angles not in quadrant one, we can simply use what's called a reference angle. Every single one of these angles corresponds to an angle in quadrant one that we already know absolutely everything about allowing us to find our trig values really quickly and easily for all of these other angles. But before we even think about trig values, we first need to know how to even find a reference angle. So here I'm going to show you how to do just that. Let's go ahead and jump right in here. Now, like I said, for angles that are not in quadrant one, they all correspond back to angles that are in quadrant one. So every single one of these angles here has a reference angle of 3045 or 60 degree. Now, how exactly do we find that reference angle? Well, let's take a look here at our 150 degree angle. We know that this angle measures 150 degrees away from our zero degree measure there. But what if instead we were to measure this angle from the other side of the X axis here at 180 degrees? Well, this angle formed with this side of the x axis from 150 degrees to 180 degrees is only 30 degrees. And if I were to draw this triangle out, we actually see that it forms the exact same triangle as our 30 degree angle measure in quadrant one just flipped over to the other side. So our 150 degree angle has a reference angle of 30 degrees. And this is how it's going to work for all of these angles. We're always going to measure them to the nearest part of the X axis and write this angle measure as a positive number. So this angle formed here is a positive 30 degrees and that represents its reference angle. Now, let's take a look at our remaining angles in this quadrant two, starting with 135 degrees. Now, we want to measure this again to the nearest part of the x axis, which still happens to be this 180 degrees over here. Now, from 180 to 135 degrees. This measures 45 degrees away. So it has a reference angle of 45 degrees which again, we can see that it forms the same exact triangle just flipped in the opposite direction. Now, our final angle and quadrant two is this 120 degrees which when I take a look at this angle and measure it to the nearest part of the X axis. Again, still 180 degrees. This is 60 degrees away from 120 down to 80. So this tells me that 120 degrees has a reference angle of 60 degrees. And again, we can take a look at that triangle and verify that those are the exact same triangle just flipped in the other direction. Now, we want take a look at our remaining two quadrants quadrant three and quadrant four. And we're going to do the exact same thing here measuring to the nearest part of the X axis to determine that reference angle. Now, remember we're going to write these as a positive number because it doesn't matter what quadrant is, this is always going to have a positive reference angle. So let's first take a look at 210 degrees and 330 degrees. Now, for 210 degrees here, looking at its nearest x axis here, the nearest X axis is still that 180 degrees. And I see that it does measure 30 degrees away from that. So it has a reference angle of 30 degrees. Now, looking at 330 degrees, it's closest X axis is coming around to a full rotation at 360 degrees, which is also 30 degrees away. So it too has a reference angle of 30 degrees. Now, let's take a look at two more angles here, we have 225 degrees and 315 degrees. Now, again, measuring to their nearest point on the X axis here, looking at 225 and 315 these are directly in the center angle wise in those quadrants. The same way that 45 degrees is. So when we look at that angle measure, we see that those are both 45 degrees away from their nearest X axis. So they both have a reference angle of 45 degrees. Now, for two remaining angles here, we have 240 degrees and 300 degrees. Now, again, measuring to the nearest part of the X axis as we have every other time we see that these are both 60 degrees away from their nearest X axis respectively. So they each have a reference angle of 60 degrees. Now, you might see that I've color coded these here and this is also a way that's going to allow us to kind of remember what all of our reference angles are. So whenever I'm remembering reference angles, I like to think of them as forming an X. When I look at all of my 30 degree angles or my 30 degree, my angles that have a reference angle of 30 degrees, they form a perfect X with that reference angle 30. Now, the same thing is true of our 45 degree angles looking here, all of these angles that have that reference angle of 45 form a perfect X. Now, the same thing is true of all of my angles that have a reference angle of 60 forming a taller thinner X. Now, the other way that you can think about this is by looking at these radiant angle measures because all of the ones that have a corresponding reference angle have the exact same denominator. So all of our 30 degree reference angles have a denominator of six. When we look at this here, then for 45 degrees, they all have a denominator of four when looking at that radiant angle measure. And finally, all of my 60 degree reference angles have a denominator of three. Now, that's a couple of different ways that you can choose to think about that. And now we know what we need to know about finding reference angles. So thanks for watching and I'll see you in the next one.

2

Problem

Problem

Identify the reference angle of each given angle.

120°

A

30°

B

45°

C

60°

3

Problem

Problem

Identify the reference angle of each given angle.

$\frac{7\pi}{4}$ rad

A

$\frac{\pi}{6}$

B

$\frac{\pi}{4}$

C

$\frac{\pi}{3}$

4

Problem

Problem

Identify the reference angle of each given angle.

210°

A

30°

B

45°

C

60°

5

concept

Trig Values in Quadrants II, III, & IV

Video duration:

5m

Play a video:

Hey, everyone, we now know all of the trig values of our three common angles in quadrant one. And we also know that all of our other angles have a reference angle equal to one of those common angles. Now, all that's left to find is the trig values of all of these angles and learning that many more new trig values doesn't not sound like fun. But luckily, you don't have to do that at all because the trig values of all of these angles are the exact same as those of their reference angle with just one tiny difference that I'm going to walk you through right now. So let's go ahead and get started. Now, I mentioned one tiny difference and that difference is the sign. So the sign cosine and tangent of each of these angles is the exact same value as that of the reference angle, but just with a different sign depending on what quadrant our angle is located in. So let's go ahead and just come right down here and look at our unit circle. So in quadrant one, we know all of these trick values, the cosine sine and tangent of each of these angles. And we see that being in quadrant one, these are all positive values. But what about as we move over to quadrant two, let's specifically take a look at our 150 degree angle here. Now remember that this has a reference angle equal to 30 degrees because that's the angle it makes with the nearest X axis. So with reference angle of 30 degrees, we can further visualize this by seeing that they form the exact same triangle just flipped. So it makes sense that the base and height of these triangles would be the exact same which are cosine and sine values. So we can take the cosine, sine and tangent here are trig values and just copy them over here because they're the exact same value. Now doing that, I end up with route 3/2 for that cosine or that X value one half for that Y value that S and root 3/3 for tangent. Now, the only thing that remains to consider here is the sign of each of these values because we're no longer located in quadrant one. Now looking at our relationship to the X and Y axis, I see that my X values over here should be negative whereas my Y values remain positive. So I want to reflect that in the stric values. So my X value is negative, negative root the over two, my Y value is positive. Now, when looking at the tangent, we just consider how we actually find the tangent, right. So the tangent is the sign over the cosine or Y over X. So knowing that our Y value is positive and our X value is negative, if I were divi to divide a positive value by a negative value, that would give me a negative value. So here, my tangent should also be negative. Now, looking at these values altogether, only one of them is positive. And this will be true for all of our trig values in quadrant two, only the S value or Y value here is going to be positive for all trig values in quadrant two. Now let's move on to quadrant three and specifically 225 degrees. Now I know that the reference angle of 225 degrees is 45 degrees in quadrant one. So I can take my trig values here and simply copy them over here for the trig values of 225 degrees. So here I can copy these in with the cosine and sine both being route 2/2, my X and Y values and then the tangent as one. Now, all that we need to consider here is remember the sign. So based on where we are in the X and Y axis, we see that these are both, these both should be negative values. My X and my Y value are both negative. Now thinking about the tangent in order to find the tangent, I would be dividing a negative by a negative, which would actually give me a positive value. So my tangent here in quadrant three remains positive. Now, looking at these values altogether, the only one that's positive is tangent, which will of course be true of all of my trig values in that quadrant. So here in quadrant three, the tangent will always be positive and it's the only positive one. Now, finally, let's take a look over here at quadrant four. Now, in quadrant four, we're going to look specifically at 300 degrees. Now we know that 300 degrees has a reference angle of 60. So I can go ahead and take my 60 degree trig values and copy them right down here for the trig values of 300 degrees. Now doing that, my cosine and my si get one half and a root 3/2, literally just copying those values. Then for the tangent route three, now we need to consider, of course, the sign. Now the sign here looking in this quadrant quadrant four, we know that we are in the negative Y values and positive X value. So we need to reflect that on these trig values. So negative Y values, this is negative route 3/2. And then for our tangent, we need to consider what we're doing. Of course, here we would be dividing a negative by a positive which would give us a negative value. So our tangent here will all so be negative. Now, finally looking at all three of these trig values as a whole, the only one that's positive is this cosine, this X value one half. So here in quadrant four, the cosine will always be positive. OK, we've looked at all of our quadrants here and we've seen the sign difference in each of them. All of our trig values are the same just with a little variation in sign depending on our location and on the unit circle. Now, how can we remember this? We can remember this using a pneumonic device using the first letter of each function that's positive. So in quadrant one, all of them are positive quadrant two, the sign is positive quadrant three, the tangent and quadrant for the cosine a stc. All students take calculus. Now, this might not always be a true statement, but it's one that can help us remember which trig function is positive in our unit circle. So now we have all of the information we need in order to completely fill in an entirely blank unit circle. So let's continue getting some more practice together. Thanks for watching and I'll see you in the next one.

6

example

Example 1

Video duration:

3m

Play a video:

Hey, everyone in this problem, we're asked to use reference Eagles to complete the missing trig values in quadrants 23 and four. And now that we know that all of these trig values are going to be the exact same as those in quadrant one with just varying sign based on quadrant, we can do this really quickly. So remember our mnemonic device here of how we remember which function is positive in each quadrant, all students take count is telling us the first letter of each function that's positive within each quadrant. Now let's take first take a look at quadrant two and looking at 135 degrees and 120 degrees. Now, these have reference angles of 45 degrees and 60 degrees respectively. So we could just go ahead and copy those trick values over to this quadrant. So for 45 or for 135 degrees, I have route 2/2, route 2/2 and one. Then for 120 degrees, I have one half root 3/2 and the square root of three for that tangent. Now, in this quadrant, only the sign is positive based on that Pneumonic device. So that means that both my cosine and tangent need to be negative here. So my X values of both of these should be negative as well as this tangent here on the end. Now, we're done with quadrant two. Let's move on to quadrant three down here. Now, looking at quadrant three, I have 210 degrees and 240 degrees as those missing trig values. So looking back up to quadrant one, I know that 210 degrees has a reference angle of 30 degrees and 240 degrees has a reference angle of 60 degrees. So I can copy those trig values over. So for 210 degrees, I have the square root of 3/2 for my cosine one half for my sign and the square of 3/3 for that tangent. Then for 240 degrees or four pi over three radiance, I have one half for the cosine root 3/2 for the sine and a root three for that tangent. Now, in this quadrant only the tangent is positive. So that means both my sine and cosine need to be negative. So X and Y here are both a negative when looking at these coordinates. Now finally, let's take a look at quadrant four over here. Now looking in this quadrant, my missing information is for 333 115 degrees which have reference angles of 30 degrees and 45 degrees respectively. So let's go ahead and copy those trig values over here in this quadrant four. So for 330 degrees, I have the square root of 3/2, 1 half and the square root of 3/3. Then for that 315 degrees, I have the square root of 2/2 square root 2/2. And finally, one for that last tangent value. Now in quadrant four, only the cosine is positive here. So that tells me that I need to go ahead and make both my sign and my tangent negative. So my sign value and my tangent negative in both of these angles. Now we have completely filled in all of our information for the unit circle. Now continue to practice this on your own and check in with me as needed. Thanks for watching and I'll see you in the next one.

7

Problem

Problem

Identify what angle, $\theta$ , satisfies the following conditions.

$\sin\theta=\frac12$; $\tan\theta$ < $0$

A

30°

B

150°

C

60°

D

300°

8

Problem

Problem

Identify what angle, $\theta$ , satisfies the following conditions.

Identify what angle, $\theta$ , satisfies the following conditions.

$\tan\theta=-1$; $\cos\theta$ > $0$

A

45°

B

135°

C

315°

D

330°

10

example

Example 2

Video duration:

8m

Play a video:

In this problem, we're given a completely blank unit circle and asked to fill in all of the missing information. Now, this is a specific type of problem that you'll get better at with repetition. And it's definitely something that you should try on your own before jumping back in with me. Now, here, I'm going to walk you through my thought process and filling in the unit circle and what works best for me might not be what works best for you. So of course, try this on your own and figure out exactly what method works best for you for filling in the entire unit circle. Let's go ahead and get started here. Now, looking at our unit circle, the first thing that I like to do is fill in all of the information for my four quadrant angles because to me, it's the simplest information on here. Now, looking at this first angle measure I have zero degrees or zero radiance and I'm gonna go ahead and fill in all of my angle measures for my three other quadrant angles up here. I have 90 degrees, 180 degrees, 270 degrees and remember coming back around to a full rotation is 360 degrees. Now, off times a blank unit circle will only have a one blank there. But it's important to remember that this is also a full rotation around. Now, let's also fill in those radiant angle measures. We have pi over two radiance, pi radiance and three pi over two radiance. Now we have all those angle measures filled in in both degrees and radiance. So let's consider our trig values of our quadrant angles as well. Now remember on the coordinate system, we know that this point is located right here at 10. And then we find our tangent value by simply dividing Y over. So doing that here, I have zero over one which gives me a value of zero. Then up here for 90 degrees, this is located at the 0.01 and dividing Y over X here gives me one over zero and undefined value. Now, over here with pi radiance, we're at negative 10 and we again get zero for a tangent and then down here at 270 degrees, we're at zero and negative one, which is also an undefined tangent value. OK, we have all of the information for our quadrant angles. So now let's focus in on all of the other angle measures for all of our other quadrants. Now, let's start with quadrant one here because these are likely the easiest for you to remember as they are for me. So here we're looking at our three common angles, we have 30 degrees, 45 degrees and 60 degrees. Now, I'm also gonna go ahead and fill in my radiant angle measures. Here in this first quadrant, we have pi over six pi over four and pi over three. Now here, let's go ahead and focus on all of our reference angles. Now, all of these angles here have a reference angle of 30. So that's gonna really help me to figure out what these angle measures are. So let's start here in quadrant two. Now in quadrant two, I look at this reference angle in reference to the nearest part of the X axis which is at 180 degrees. So this is 30 degrees away from that 180 which tells me I can find this angle measure by simply taking a 180 minus 30 degrees. Now this gives me an angle measure of 150 degrees. Now, for that radiant angle measure, I can actually do the same exact thing, but just with pi this is pi over six radiant away. So I take pi minus pi over six here and that gives me my gradient angle measure of five pi over six. Now this will work for any angle in this quadrant because we can just do that for any angle, we just subtract our angle from 180. And from pi now let's look in our quadrant three here and do the same exact thing. Now, this is in the opposite direction of 180. So here I can actually take 180 add my angle to it instead to get that angle measure. Now, this is 30 degrees away because we're still looking at our 30 degree reference angle here. So if I take 180 I add 30 degrees, that's going to give me an angle measure of 210 degrees. Now I can do the same thing with pi. Of course, I add pi to my angle here which is pi over six. So pi plus pi over six gives me an angle measure of seven pi over six. Now let's move on to quadrant four. Now we look at this in reference to our final angle, our full rotation of 360 degrees. And this is of course still a 30 degree reference angle. So here I can take 360 degrees and I can subtract my angle here. So 360 minus theta my angle. Now, here this is 30 degrees. So 360 minus 30 it gives me 330 degrees. Now, of course, this is also two pi radiance. So I can also take two pi and subtract my angle in radiance to get the same to get my radiant angle measure here. So here two pi minus pi over six will give me 11 pi over six. Now, looking at all of these radiant angle measures, you can also choose to memorize these sort of patterns that happen with your radiant angle measures. So looking at all of these, I have pi over 65 pi over 67 pi over six and 11 pi over six. Now four angles that have the same reference angle, they always have the same denominator in that radiant angle. These all have a denominator of six and they count from 15 7-Eleven. So you can remember that pattern if it's easier for you to memorize here. Now, here looking at this, I can apply the same line of thinking to all of my other reference angles. So I want you to go ahead and try that on your own and then check back in with me. OK. Now that we have all of those angle measures in there, we can move on to trig values. Now, let's start in quadrant one because we have some memory tools that will help us fill this in rather easily. Now remember in quadrant one, we always start the same way regardless of how we choose to memorize this, we always start with the square root of two for every single one of these values. So I can go ahead and fill that in for every trig trig value in quadrant one. Now, once I do that, I can go ahead and use a memory tool. Now, here, I'm going to use the 123 memory tool, but you can of course, use whatever works for you. So here using the 123 memory memory tool, I'm going to start here with this value and I'm going to count 123 in the clockwise direction for those X values and then 123 back in the counter clockwise direction for those Y values. So these are my trig values in quadrant one. Now, we can of course simplify these square roots of one because they're just one. Now, from here, we want to fill in our tangent values. Now, you can choose to memorize your tangent values. But it's easier for me to just remember that this is Y over X. So I can take these values that I already have and simply divide them now because they have the same denominator, we're effectively just dividing those numerators. So here in this first one, I divide my numerators Y over X one over the square root of three. Now this is a correct value. But remember it's gonna be better if we rationalize our denominator here. So this will give me the square root of 3/3 for the tangent of 30 degrees. Now for 45 degrees, if I take Y over X, those numerators, I get a value of one and then for 60 degrees, I get the square root of 3/1 which is just the square root of three. Now, we have all of our trig values in quadrant one, but we have these three other quadrants. Now, remember all our trig values are the exact same as those of their reference angles and we only have to worry about the sign. So from here, I'm just going to go ahead and fill in all of these trig values, copying them over from quadrant one to their respective reference angle in every single quadrant. And then we can worry about the sign after. So I'm going to go ahead and fill all of those values in now. OK. All of our trig values are copied over to quadrants 23 and four. But remember the sign of these values may be different depending on what quadrant they're in. So that's where we consider it the sign. Now, remember a mnemonic device to remember the signs of these values. All students take calculus telling us the first letter of whatever trig function is positive in that quadrant. So in quadrant two, we have s that tells us that only the sign is positive. So only my Y values here root 3/2 square root of 2/2 and one half should be positive. So all of my other values are negative and I can go ahead and fill that in for all of my other values, putting a negative sign on each of them. Now, we can do the same exact thing in quadrants three and four. Now in quadrant three, only the tangent is positive. And in quadrant four, only the cosine is positive. So all of my other trick values must be negative. I'm going to go ahead and copy those negative sign. And now based on that and then we'll rejoin together. OK? You should have all of your trig values filled in. Now with negative signs on the correct values based on our pneumonic device. All students take calculus. So take a chance to double check all of your values here with mine. Now, we have completely filled in all of the missing information on our unit circle. Now again, this is a problem that gets easier with repetition. So feel free to try this as many times as you need until you have it down. Thanks for watching. And of course, let me know if you have questions.

11

concept

Coterminal Angles on the Unit Circle

Video duration:

3m

Play a video:

Hey, everyone, if I gave you a unit circle and I asked you to find the sign of 30 degrees, you'd probably locate 30 degrees on your unit circle and simply identify your white value. But what if instead I asked you to find the sign of 390 degrees. It doesn't sound quite so simple. But if we look at our circle here and we go a full rotation around from 0 to 360 degrees and then just go an additional 30 degrees. I'm actually right here at 390 degrees. So the values are going to be the exact same as those of 30 degrees an angle that I already know. And that's because these angles are co terminal. So their trig values are going to be the exact same. Now, the first time that you see an angle that's either really large or maybe even negative, it might seem a little bit strange. But here, I'm going to walk you through exactly how to identify a co terminal angle on the unit circle and use that to really quickly and easily identify your trick values. So let's go ahead and get started here. Now, you might remember co terminal angles more formally as an angle with the same terminal side. And you might also remember finding them by adding or subtracting multiples of 360 degrees or two pi radiance. So with my 390 degree angle, if I were to subtract a full rotation around a circle of 360 degrees, I would end up with my co terminal angle of 30 degrees. Now, we're going to do basically the same thing here, but it's going to be even easier to visit lies on our unit circle because we already have all of these angles here. So we can do this without having to set up any algebraic equation and just visually finding these angles. So let's go ahead and jump right into this first example and find the tangent of three pi. Now the tangent of three pi, the first thing we want to do here is find the co terminal angle of three pi so that we can easily find our trig value that's already on our unit circle. So let's consider this by looking at our unit circle if I go around my circle once from 0 to 2 pi radiance, and then I go another additional pi radiance, I've really gone three pi radiance around my circle. But I've ended right back here at pi which is my co terminal angle. So the co terminal angle of three pi is simply pi. So to find the tangent, I just need to look at the tangent value of pi here, which happens to be zero. So the tangent of three pi is equal to zero. And we're done here. Now let's move on to our next example here here, we're asked to find the cosine of negative pi over four. Now remember for a negative angle that just tells us that we're going clockwise around our circle rather than counterclockwise. So let's again visualize this on our unit circle here. Now, if I start at zero and I go negative pi over four radiance, so this way, clockwise, negative pi over four, I'm gonna end up here at seven pi over four radiance. So the co terminal angle of negative pi over four is seven pi over four. Now that I know that I can identify the cosine by just finding the cosine of that co terminal angle seven pi over four, which is of course my X value here. So the cosine of negative pi over four is route 2/2. And I'm good to go here. Now, we have one final example, the sign of 390 degrees. So we want to go ahead and identify this on our unit circle. Now, you might remember this from earlier, but let's just visualize this one more time. So 390 degrees, if I go around my circle once that's 360 degrees and if I go another 30 degrees that puts me at 390 degrees. So the co terminal angle here is of course, 30 degrees. Now to find the sign of this angle, we're just going to look at the sign of 30 degrees, which is of course, our Y value in this case, one half. So the sign of 390 degrees is one half. Now that we know how to find trig values using co terminal angles. Let's get some more practice together. Thanks for watching and let me know if you have questions.

12

Problem

Problem

For each expression, identify which coterminal angle to use & determine the exact value of the expression.

$\sin\frac{7\pi}{3}$

A

$\frac12$

B

$2$

C

$\frac{\sqrt3}{2}$

D

$\frac{2\sqrt3}{3}$

13

Problem

Problem

For each expression, identify which coterminal angle to use & determine the exact value of the expression.

$\tan765\degree$

A

$-1$

B

$1$

C

$0$

D

undefined

14

Problem

Problem

For each expression, identify which coterminal angle to use & determine the exact value of the expression.