Reference Angles - Online Tutor, Practice Problems & Exam Prep

Hey, everyone. We now know all of our trig values of our 3 common angles in quadrant 1. But the unit circle has 3 other quadrants, all of which we'll need to know trig values in. 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 1, we can simply use what's called a reference angle. Every single one of these angles corresponds to an angle in quadrant 1 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. As I said, for angles that are not in quadrant 1, they all correspond back to angles that are in quadrant 1. So every single one of these angles here has a reference angle of 30, 45, or 60 degrees.

Lets take a look here at our 150 degree angle. We know that this angle measures 150 degrees away from our 0 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, our 150 degree angle has a reference angle of 30 degrees. 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 2, starting with 135 degrees. 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. Our final angle in quadrant 2 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. So this tells me that 120 degrees has a reference angle of 60 degrees. We can take a look at that triangle and verify that those are the exact same triangle just flipped in the other direction.

Continuing to our remaining two quadrants, quadrant 3 and quadrant 4, we're going to do the exact same thing here, measuring to the nearest part of the x-axis to determine that reference angle. Remember, we're going to write these as a positive number because it doesn't matter what quadrant it is. Let's first take a look at 210 degrees and 330 degrees. For 210 degrees, looking at its nearest x-axis, the nearest x-axis is still that 180 degrees. And I see that it measures 30 degrees away from that. So it has a reference angle of 30 degrees.

Looking at 330 degrees, its 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. Next, we have 225 degrees and 315 degrees. Measuring to their nearest point on the x-axis, 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.

Our two remaining angles here, 240 degrees and 300 degrees. 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.

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 angles that have a reference angle of 30 degrees, they form a perfect x with that reference angle 30. 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. The same thing is true of all of my angles that have a reference angle of 60 forming a taller, thinner x. Another way that you can think about this is by looking at these radian 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 6 when we look at this here, then for 45 degrees, they all have a denominator of 4 when looking at that radian angle measure. And finally, all of my 60-degree reference angles have a denominator of 3. 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.

Hey, everyone. We now know all of the trig values of our 3 common angles in quadrant 1. 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 does 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 sine. So the sine, cosine, and tangent of each of these angles are 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 1, we know all of these trig values, the cosine, sine, and tangent of each of these angles. And we see that being in quadrant 1, these are all positive values. But what about as we move over to quadrant 2? 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 that 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, our trig values, and just copy them over here because they're the exact same value. Now doing that, I end up with 3/2 for that cosine or that x value, one-half for that y value, that sine, and 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 1. Now looking at our relationship to the x and y axes, I see that my x values over here should be negative, whereas my y values remain positive. So I want to reflect that in these trig values. So my x value is negative root 3 over 2. 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 sine over the cosine or y over x. So knowing that our y value is positive and our x value is negative, if I were 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 2. Only the sine value, our y value here, is going to be positive for all trig values in quadrant 2. Now let's move on to quadrant 3 and specifically 225 degrees. Now I know that the reference angle of 225 degrees is 45 degrees in quadrant 1, 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 root 2 over 2, my x and y values, and then the tangent as 1. 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-positive value. So my tangent here in quadrant 3 remains positive. Now looking at these values altogether, the only one that's positive is tangent, which will, of course, be true for all of my trig values in that quadrant 3. So here in quadrant 3, the tangent will always be positive, and it's the only positive one.

Now finally, let's take a look over here at quadrant 4. Now in quadrant 4, we're gonna 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 sine, I get one-half and root 3 over 2, literally just copying those values, then for the tangent, root 3. Now we need to consider, of course, the sign. Now the sign here, looking in this quadrant, quadrant 4, we know that we are in the negative y-values and positive x-values, so we need to reflect that on these trig values. So negative y-values, this negative root 3 over 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 me a negative value. So our tangent here will also be negative. Now finally, looking at all 3 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 4, the cosine will always be positive.

Okay. 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 on the unit circle. Now how can we remember this? We can remember this using a mnemonic device first letter of each function that's positive. So in quadrant 1, all of them are positive. Quadrant 2, the sine is positive. Quadrant 3, the tangent. And quadrant 4, the cosine, AST sign, ASTC. 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.

Hey, everyone. In this problem, we're asked to use reference angles to complete the missing trig values in quadrants 2, 3, and 4. And now that we know that all of these trig values are going to be the exact same as those in quadrant 1 with just varying sign based on the 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 calculus, telling us the first letter of each function that's positive within each quadrant. Now let's first take a look at quadrant 2, 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 trig values over to this quadrant. So for 135 degrees, I have √2/2, √2/2, and 1. Then for 120 degrees, I have 1/2, √3/2, and the square root of 3 for that tangent. Now in this quadrant, only the sine is positive based on that mnemonic 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 2. Let's move on to quadrant 3 down here. Now looking at quadrant 3, I have 210 degrees and 240 degrees as those missing trig values. So looking back up to quadrant 1, 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 √3/2 for my cosine, 1/2 for my sine, and √3/3 for that tangent. Then for 240 degrees or 4π/3 radians, I have one half for the cosine, √3/2 for the sine, and a root 3 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 negative when looking at these coordinates. Now finally, let's take a look at quadrant 4 over here. Now looking in this quadrant, my missing information is for 330 and 315 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 4. So for 330 degrees, I have √3/2, 1/2, and √3/3. Then for 315 degrees, I have √2/2, √2/2, and finally, one for that last tangent value. Now in quadrant 4, only the cosine is positive here, so that tells me that I need to go ahead and make both my sine and my tangent negative. So my sine 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.

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. It's definitely something that you should try on your own before jumping back in with me. Here, I'm going to walk you through my thought process in 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.

Looking at our unit circle, the first thing that I like to do is fill in all of the information for my 4 quadrantal angles because to me it's the simplest information on here. Now looking at this first angle measure, I have 0 degrees or 0 radians and I'm going to go ahead and fill in all of my angle measures for my 3 other quadrantal angles. Up here I have 90 degrees, 180 degrees, 270 degrees, and remember coming back around to a full rotation is 360 degrees. Oftentimes a blank unit circle will only have one blank there but it's important to remember that this is also a full rotation around. Let's also fill in those radian angle measures. We have π2 radians, π radians, and 3π2 radians. Now we have all those angle measures filled in in both degrees and radians, so let's consider our trig values of our quadrantal angles as well.

Remember on the coordinate system we know that this point is located right here at (1,0) and then we find our tangent value by simply dividing y over x. So doing that here I have 0 over 1 which gives me a value of 0. Then up here for 90 degrees, this is located at the point (0,1) and dividing y over x here gives me 1 over 0, an undefined value. Now over here with π radians, we're at (-1,0) and we again get 0 for our tangent. And then down here at 270 degrees we're at (0,-1) which is also an undefined tangent value. Okay, we have all of the information for our 4 quadrantal angles. So now let's focus on all of the other angle measures for all of our other quadrants.

Let's start with quadrant 1 here because these are likely the easiest for you to remember as they are for me. Here we're looking at our 3 common angles. We have 30 degrees, 45 degrees, and 60 degrees. I'm also going to go ahead and fill in my radian angle measures here. In this first quadrant, we have π6, π4, and π3. Now here, let's go ahead and focus on all of our reference angles. All of these angles here have a reference angle of 30, so that's going to really help me to figure out what these angle measures are.

Let's start here in quadrant 2. Now in quadrant 2, 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 180 minus 30 degrees. This gives me an angle measure of 150 degrees. For that radian angle measure, I can actually do the same exact thing but just with pi. This is π6 radians away, so I take π minus π6 here and that gives me my radian angle measure of 5π6. 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 3 here and do the same exact thing. This is in the opposite direction of 180, so here I can actually take 180 and add my angle to it instead to get that angle measure. This is 30 degrees away because we're still looking at our 30 degree reference angle here. So if I take 180 and I add 30 degrees, that's going to give me an angle measure of 210 degrees. I can do the same thing with π of course. I add π to my angle here which is π6. So π + π6 gives me an angle measure of 7π6.

Now let's move on to quadrant 4. Here 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. Here this is 30 degrees, so 360 minus 30 gives me 330 degrees. This is also 2π radians, so I can also take 2π and subtract my angle in radians to get the same to get my radian angle measure here. So here, 2π minus π6 will give me 11π6x.

Hey everyone! If I gave you a unit circle and I asked you to find the sine of 30 degrees, you would probably locate 30 degrees on your unit circle and simply identify your y value. What if instead I asked you to find the sine 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 trig 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 coterminal, 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 coterminal angle on the unit circle and use that to really quickly and easily identify your trig values. So let's go ahead and get started here. Now you might remember coterminal 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 2π radians. 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 coterminal angle of 30 degrees.

Now we're going to do basically the same thing here, but it's going to be even easier to visualize 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 3π. Now the tangent of 3π. The first thing we want to do here is find the coterminal angle of 3π 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 2π radians and then I go another additional π radians, I've really gone 3π radians around my circle, but I've ended right back here at π, which is my coterminal angle. So the coterminal angle of 3π π is simply π π. So to find the tangent, I just need to look at the tangent value of π π here, which happens to be 0.

The tangent of 3π is equal to 0, 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 π/4. 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 0 and I go negative π/4 radians, so this way, clockwise, negative π/4, I'm gonna end up here at 7π/4 radians. So the coterminal angle of negative π/4 is 7π/4.

Now that I know that, I can identify the cosine by just finding the cosine of that coterminal angle 7π/4, which is of course my x value here. So the cosine of negative π/4 is 22 and I'm good to go here.

Now we have one final example, the sine 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 coterminal angle here is, of course, 30 degrees. Now to find the sine of this angle, we're just gonna look at the sine of 30 degrees which is of course our y value, in this case, 1/2. So the sine of 390 degrees is 1/2.

Now that we know how to find trig values using coterminal angles, let's get some more practice together. Thanks for watching and let me know if you have questions.