Angles: Videos & Practice Problems

Angles are measured in degrees and radians, with a full circle equating to 360 degrees or 2π radians. To convert between these units, use the formulas: $\frac{\pi }{180}for\; degrees\; to\; radians,\; and$ \frac{180}{\pi }for\; radians\; to\; degrees.\; Angles\; can\; also\; be\; coterminal,\; meaning\; they\; share\; the\; same\; terminal\; side,\; found\; by\; adding\; or\; subtracting\; multiples\; of\; 360\; degrees.$$

Drawing Angles in Standard Position

Everyone. So in a previous video where we discussed the basics of triangles, we said one way to classify a triangle was based on its angle, and an angle is just where two of the sides of a triangle meet. Well, a lot of times we're going to be working with angles. We'll need to work with them not in a shape or floating around in space, but instead on an x and y coordinate system like this. Angles are one of the most important things that you'll need, not just for this course, but you'll also be using them regularly in other math courses.

I want to make sure you have a great understanding of them. So I'm going to introduce you to angles. I'm going to show you how to draw them with some conceptual things you'll need to know, and then we'll do some examples together. Let's get started here. So the first I want to talk about is the definition of an angle.

Your textbooks won't have a great definition for this, but it's basically just the gap or the space that you create between two line segments or sides of a triangle. A lot of times when we work on the x y coordinate system, we don't even care about that third side of the triangle. But the angle is really just sort of this gap that you make here in between these two line segments. Alright? Now the bigger the gap, the bigger the angle.

And we came up with a number system for this a long time ago, like thousands of years ago, where basically you measure in degrees from zero all the way to 360 as you go around the x y coordinate system. The ancient Babylonians a long time ago decided that the x axis over here would be zero, and then as you go around, the angle gets bigger. If you go around a full circle over here, you've gone a full 360 degrees. That's probably another you're familiar with, which actually cuts up our x and y axis into increments of 90. Right?

You go quarter of the circle, that's 90. Another quarter of the circle, that's another 90, so it'd be one eighty. Another quarter of the circle, this would be two seventy. And then all the way back around will be 360, and then you would just start all the way over again. Alright.

So then what would something like 60 degrees look like? Let's just get right into our first example. If these things are 90, one 80, and two seventy, what would 60 degrees look like? Well, first of all, how do you even draw what an angle is? How do I know which of the sides to draw?

Can I draw an angle that kind of looks like this with those two sides? But what we do is a long time ago, we decided that the way we're going to draw angles is from the initial side. The initial side is going to be along the positive x axis. So in other words, it's going to be this side over here, and you're going to draw from that to the terminal side. The terminal side is usually going to be the one that's not on the x axis.

Alright? So that is the terminal side. Whenever you have an angle that's drawn like this and all of your angles are going to be drawn like this, your textbooks refer to that as an angle drawn in standard position. In fact, most of the time, you're not even going to see this initial side. You'll just see something that looks like this.

That's how your angles will be shown. Alright? So that's how you draw an angle. So then how do you get to 60 degrees? Well, basically, you can use your axes as guides for this.

Right? If I have an angle that sweeps out like this and I go all the way around, whoops, an angle that looks like this, that would be 90 degrees. Right? So I want something that's going to be a little bit less than 90. Alright?

If I draw something that looks halfway between 90, that's going to be 45 degrees halfway between 90. So I need something that looks a little bit sort of more steep than that. In fact, it's actually going to look something like exactly what I've drawn over here with this line. If you drew it exactly like this, this is about what 60 degrees is going to look like. Now, again, this has to you know, it doesn't have to be perfect because it's just a sketch, but but I have a little protractor here that kind of helps me sort of gauge these angles.

And if you draw this, this is going to be about, let's about, like, fifty degrees, which is close enough. Right? That's what I've got there. Alright. So that's how you draw 60 degrees.

Let's move on to the next one. The next one I'm going to draw in purple over here. How do you draw 150 degrees? Well, again, the initial side is going to be the same. It's always going to be along that positive x axis.

And what I'm going to do here is I'm going to start to draw a little angle that goes all the way around in this direction. Alright. So notice here how this goes from zero to 90. A 150 is going to be a little bit more than 90. But if you draw all the way to the left, that's going to be a 180, which is going to be too much.

So it's going to have to be somewhere here in this quadrant, and it's going to be close to one eighty, not exactly close to 90. So it's going to look something like this. Alright. If you measure this angle all the way from the positive x axis like this, that's going to be about 150 degrees. Alright.

So we've got those two that are done. So pretty straightforward there. Now one of the things you might have noticed here is that both of these examples have been positive angles. We've always been drawing them up like this from the positive x axis in the counterclockwise direction. This is just a convention that we came up a long time ago with, which is that we're going to draw positive angles in the counterclockwise direction against the direction of the clock.

That just means that if you have a negative angle, like for example, negative 60 degrees, you're going to go the other way. You're going to draw those in the clockwise direction. Alright? So how would I draw negative 60? Well, positive 60 would be if I went up from zero like this and drew an angle like that.

Negative 60 would be in the opposite direction. I'd have to go down like this in this direction, and if I go clockwise down from zero, that's going to be negative 60 degrees. That's one way I like to remember this. Positive angles go up from zero, Clockwise negative angles will go down from zero. That's one way I like to remember this.

That's really all there is to it. That's how you sketch angles in standard position. Alright? So one last thing I want to mention here is you may be asked to sort of classify types of angles, and it's actually very similar to how we did this for triangles. So, the three words we use for this are going to be acutes, obtuse, and rights, exactly like what we use for triangles.

And basically, those things have to do with what they are relative to ninety. Acute angles are going to be less than 90 degrees. So for example, this is an acute angle, that example a that we worked with. Example b, which is 150 degrees, that's going to be something that's bigger than 90, so that's an obtuse angle. And a right angle is exactly equal to 90 degrees.

Alright? There's something really quick that you might be asked for. That's it for this one, folks. Thanks for watching, and let me know if you have any questions.

What is the approximate measure of the angle shown below? Choose the most reasonable answer.

• A. $60°$
• B. $150\degree$
• C. $240\degree$
• D. $300\degree$

Answer: (C)

Drawing Angles in Standard Position Example 1

Everyone, let's use what we know about angles to go ahead and work out this example. We're going to draw and sketch each of these angles in standard position. They're just sketches, so they can kind of be rough. They don't have to be perfect.

Let's go and take a look at the first one, which is 45 degrees. What does that look like? Well, let's just use our axes as guides for this because we know this is zero, we know this is 90, this is 180, and 270. So 45 is going to be somewhere between 0 and 90. It's going to be somewhere in this first quadrant over here.

What does 45 look like? Well, if you'd actually think about it, 45 is exactly halfway between 0 and 90. That's exactly what 902 is. So if I were to grab a line and start to sweep it out towards the 90 degree angle, I would stop basically at halfway, which is almost like a perfectly diagonal line like this. That's what 45 degrees looks like.

So I’m just going to take a line, and I’m going to sort of draw it so that I'm going to have to be, like, perfectly at, like, the northeast, you know, upright corner, and I’m basically just going to be cutting these two angles in half like this. So this would be 45 degrees. Alright? That's about what that would look like. Alright.

Let’s look at the next one here, which is 210 degrees. Alright? So 210. Again, let’s use our axis as guides. This is 90, and this is 180.

210 is bigger than 180, so you’re going to start to go around in this direction over here into this third quadrant because this is going to be 270. That's too big. So it's going to be somewhere over here. Alright? So if you look sort of the halfway between 180 and 270, something like the halfway points like this, this is actually 225.

You can figure that out just by taking the difference between those. Alright? So that's actually too big as well. 210 is going to be somewhere inside of over here. It’s going to be somewhere in this area.

So let's actually use sort of use what we know about 30, to sort of draw this. Because if you look at this, the difference between 180 and 210 is 30 degrees. So what does 30 look like in the first quadrant? Well, if you sort of draw this out, it's going to look something that looks like this. Right?

That's about what a 30 degree angle looks like. So the difference between 180 and 210 is also basically what this is but in the third quadrant. So all that really means here is you can take this line at 30, You can kind of just imagine that it keeps extending out in this direction over here. I'm just going to extend the line because now that angle is going to be 30 degrees from 180 degrees.

Right? So this is going to get me 210 over here. Alright? It's 30 degrees from 180, but if you go all the way around the circle, that whole angle over there is 210 degrees. Hopefully, that made sense.

Alright, now let's look at the last one over here, which is negative 100 degrees. Alright? So remember, these first two have been both positive numbers, so we’ve been going counterclockwise. Now we’re actually going to go in the clockwise direction, the same way that the clock goes because it's a negative angle.

So because it’s rolling in the negative direction, our axis won't help us so much because we know this is 90, 180, and 270, but we don’t know what these negative angles are. Well, what happens is, if you just go around in the other direction, you sort of just flip numbers. Alright? So this is 270, but this would be about what negative 90 degrees would look like.

Remember, these are right angles. Right? This would be 180, but also it would be negative 180 if you go around in this direction, and this would be negative 270. Alright? So negative 100 would be going basically a full 90 plus a little bit extra.

So if you were going in the clockwise direction, you would go past 90 and then a little bit extra, like 10 degrees. Alright? So what happens is, I'm actually going to draw a line that looks something like this. Alright? So this is a full 90 degrees in the negative direction, but then it's a little bit extra.

So this angle, which we draw clockwise, would be negative 100 degrees. Alright? It's about what that would look like. Again, these are sketches. They don't have to be perfect, but hopefully, you got something that looks like this.

Alright, let me know if you have any questions. Thanks for watching.

Which angle is NOT a positive angle drawn in standard position?

• A.
• B.
• C.
• D.

Answer: (C)

Coterminal Angles

So up until now, we've seen how to graph angles up to 360 degrees. If I go all the way around the circle, that's a full three sixty, and 30 degrees is going to look something like this. But what if I were asked to draw an angle that is not within 360 degrees? Like this first example, 390 degrees. Well, actually, let's just go and look at the circle and actually do it.

Because really, the idea is that going around a full circle is 360 degrees. So 390 would just be if I continued going around the circle an extra 30 degrees. So 360 degrees plus an extra 30 degrees over here is going to give me 390 degrees. And that actually ends up pointing in the exact same direction as 30 degrees. These are what we call coterminal angles.

Now, problems will not always be this simple, but I'm going to show you a really quick way to find coterminal angles with an angle that you're given, especially when you have really big numbers like a thousand or something like that. Let's go ahead and get started. I'll break it down for you. So, angles are coterminal if they point in the same direction. That means that they have the same initial or terminal side.

Remember, 'initial' means it's going to be along the x-axis. We always have angles that start along the x-axis. So that means that they have the same terminal side. Coterminal means the same terminal side. Alright?

Now, basically, what we saw here is that the only difference between 390 and 30 is that 390 has gone around a full rotation one time. In other words, their difference between them is just a multiple of 360, and that's the shortcut. To find angles that are coterminal with a given angle, all you have to do is add or subtract multiples of 360. Alright? So let's take a look at our first example over here, which is 390 degrees.

The coterminal angle for that is just going to be 390 minus 360, and that's just going to be 30 degrees, which is exactly what we saw over here. That's why we got 30 degrees. So let's look at the second example. How would I get something like negative 270 degrees? Well, this angle—

This example specifically says I want an angle between 360 degrees that's coterminal. So in other words, I want a positive number. So negative 270 degrees, if I were to graph this, I'd have to go in the clockwise direction until I reach negative 270 degrees. But what I can just do really quickly here is I can just add 360.

I'm going to add 360 degrees over here, and what you're going to get is you're going to get 90 degrees. So in other words, a 90 degrees positive angle is going to be coterminal with negative 270 degrees. A 90 degrees positive looks like this, right? It's just straight up along the y-axis.

So 90 degrees and negative 270 degrees are coterminal. They both point in the same exact direction. Alright. Let's look at the last one over here, which is 1,000 degrees.

That's a really big number. We wouldn't want to have to go around the circle again and again until we actually ended up at 1,000 degrees. So what we can do is really quickly find the coterminal angle by subtracting 360. If I subtract 360 from here, I'm going to get 640 degrees. Am I done?

Do I stop? Well, remember, this is still more than 360 degrees, so I have to subtract another rotation of 360. So, this would actually be if I've gone around a full circle more than twice. And so when you do this, what you're going to get is you're going to get 280 degrees. Now that this is a number between zero and 360, we can stop here, and we can just only graph what 280 degrees is.

Alright? How do we do that? Very simple. We've done this before. This is going to be about what 270 degrees looks like, along the negative x-axis.

So 280 degrees is going to look something like this, just about 10 degrees past that. So, this is 280 degrees, or this is also 1,000 degrees. They both point in the same direction. Those are coterminal. That's really all there is to it, folks.

Thanks for watching, and I'll see you in the next one.

Coterminal Angles Example 2

Everyone, welcome back. So let's take a look at this example problem. Let's use what we know about coterminal angles to find the smallest positive angle that's coterminal with the given angles that we see over here, and we're going to sketch them in standard position. Let's take a look at the first one over here, 710 degrees.

What does it mean by the smallest positive angle? Well, we could just go on this graph here and go all the way around and then go another rotation and try to figure out where 710 is. But the easiest way to do it is to use coterminal angles. If this is bigger than 360, we just have to knock, you know, 360 off. We just subtract multiples until we get to something that is between 0 and 360, the smallest positive angle.

Alright, so what is 710? We're just going to subtract 360 because it's bigger than 360. And if you actually subtract this, what you're going to end up getting is 350 degrees. So, do we keep going?

Do we subtract another multiple? Well, no. Because this is actually less than 360 degrees. So how do I draw this angle here? This is the smallest positive angle that's coterminal with 710.

And this is really just if you went one full circle around but actually not quite. You just stopped a little bit short. Right? Because we know that if you go from zero all the way around the circle, you'll end up back at 360 degrees. So 350 would be 10 degrees short of a full rotation.

So it'd be going like this all the way around, but you'd stop just around here, and you would end up with an angle that kind of looks like this. So this angle over here is 350 degrees. Alright? And that's how you would sketch that. Let's take a look at the next one over here, which is negative 37 degrees.

Now we know negative angles, you would just draw them clockwise from the X-axis, but we actually want to figure out what the smallest positive angle is. Right? That's a negative angle. So what do we do? Well, this is less than zero, so we're just going to add 360 to this.

You add 360 to negative 37, what you'll get is 323 degrees. So, do we keep going? No. Because that's the smallest number between zero and 360. So this is the smallest positive angle that's coterminal.

How do you draw this? Well, again, just use your axis as guides. You have zero, you've got 90, 180, 270, and then back to 360 degrees. So 323 is going to be somewhere in this quadrant over here. Again, you can use the halfway point to kind of gauge where this is going to be.

I draw a line like this which is halfway between 270 and 360, which is going to be about 315, and that's actually really close to what you want to draw. So you're going to draw a line that looks something like this but maybe like a little bit nudged to the right. So it's going to look something like that. Alright? So this angle drawn from the positive X-axis is 323 degrees.

Let's take a look at the last one over here, which is negative 480. Same thing as example B. We're just going to add 360 to this. Right? So add 360, and what do you get?

You're going to get negative 120. It's still a negative angle. We want the smallest positive, so you just have to add another multiple of 360. So we're just going to add another round of 360 to this. And what you should get is 240 degrees.

That is the answer that you want. Alright? So I'm actually let me go ahead and write this over here. This is going to equal 240 degrees. That's the answer.

How do you sketch this? Well, let's just use our guides again. We've got 90, 180, 270. So and then back to 360. So 240 is going to be somewhere in this quadrant over here, and 240 is a little bit closer to 270 than it is to 180.

Right? The halfway point to this would be 225. So we want something that's a little bit more vertical like this. Alright? So let's go ahead and draw this.

This is going to be something that looks about like that. You go all the way around from the positive X-axis. That's going to be our angle. That's going to be 240 degrees. Hopefully, this made sense.

Hopefully, you got to draw some of these on your own and got something that looks really similar to mine. Thanks for watching, and I'll see you in the next one.

Converting between Degrees & Radians

Everyone. Welcome back. So as we've been talking about angles, we've gotten really used to the idea of working with degrees. If you go around a circle, full circle, you'll go a full 360 degrees. And if I asked you to draw me something like a 20, you'd be able to draw me something that looks about like this.

What we're going to look at in this video is another unit of measuring angles called radians. It's kind of like how we use inches and centimeters to measure distance. We can use degrees and radians to measure angles. Alright? Now, we'll be using radians a lot in later videos when we talk about functions and graphing, but for now, I just want you to understand the basic difference between degrees and radians, and how we draw them, and more importantly, also how we convert between them.

Let's go ahead and get started here. Alright? So a radian really is just a different unit for measuring angles based on the idea of going around the circumference of a circle. Alright? So what we saw is that if you go full three full circles, that's 360 degrees.

So if I cut up the circle into 360 tiny little pieces, each one of those things would be a degree. So this tiny little sliver over here, that's what one degree is. What a radian is, is it's basically just based on the idea of going around the circumference of a circle. If you go around the circumference of a circle, that's two pi times the radius. That's a formula for geometry.

So the whole idea of a radian is if you grab the radius of a circle and you take that distance and then start going around the circumference of a circle, that same distance but now is curved, the angle that you're making there, that little pizza slice that you've just made is one radian. And it's about 57 degrees. That's about approximate, like, a little bit of a decimal. Alright? So if you go around a full circle, that's a full two pi radians.

So a full circle is three sixty degrees and two pi radians. But usually when we work with radians, we're going to work with them in multiples or fractions of pi. So just as we cut up three sixty into four quarters, we can cut up two pi into four quarters. If you go halfway around, half of two pi is one pi radian. If you go half of that, that's going to be pi over two radians.

If you go three quarters of the way around the circle, that's going to be three pi over two radians. So that's what our sort of axis become in radian form. So, how do we go back and forth between degrees and radians? Well, it really just comes down to these two formulas over here. To convert back and forth between degrees and radians or radians and degrees, we're just going to multiply whatever angle that we have by pi over one eighty or a 80 over pi.

Alright? We're going to do a couple of these examples of this. In fact, our first example is going to be, how do we convert a 20 degrees, this angle that we have over here? What does that end up being in terms of radians? Alright?

So what we're going to do here is this is an angle that's in terms of degrees. That's theta degrees. So that means we're going to use this top equation over here. Alright? So to convert this to radians, what we're going to have to do is take our 120 degrees, and we're going to have to multiply it by something.

Alright? So I don't actually want you to memorize these formulas. All you have to do is just memorize that the two numbers involved are pi and one eighty. How do you figure out which one goes on top? It really just comes down to what unit you're trying to get rid of.

I'm trying to get rid of degrees, so that means I want degrees on the bottom. I want the one eighty to be on the bottom. Because now what happens is you'll see that degrees will cancel, and then you'll end up with pi radians being on top. Alright? So then how does this simplify?

Well, what happens is one twenty over one eighty, you can knock a zero off, and 12 over 18 just simplifies to a fraction. So what happens is you're still left with a pi over here, but now this simplifies to a fraction of two thirds. So this is two pi over three radians. Alright. So this two pi over three is exactly what a 20 degrees is.

It's just written in a different unit. These two angles here mean the exact same thing. They both mean an angle that looks like this. It's just in different units. Alright.

So that's how you convert from degrees to radians. Let's do one more example over here because now we're actually going to do the opposite. Now we're going to convert this angle that's given to us in radians, and we're going to convert it to degrees. Alright? So this is an angle that's going to be theta radians.

And so what we're going to do is we're going to convert this to theta in degrees. So, how do we do this? Well, we take what the given angle is, six pi over five, and we know we're going to have to multiply by pi over one eighty or one eighty over pi. How do we know which one? Well, it just depends on what we're trying to get rid of.

We're trying to get rid of radians and be left with degrees. So that means that radians, we want to be on the bottom and eighty degrees on top. Usually, you can tell that this is going to be the case because you want the pi to cancel, so you want to divide by pi to get rid of it. Alright? So that's a good way to sort of double-check that.

So we're going to cancel out the pi's over here. And basically, when you actually sort of multiply these numbers out, what you're going to get here is you're going to get eighty degrees divided by divided by five. Alright? And if you work this out, what you'll actually end up getting here is you'll end up getting 216 degrees. Alright?

So that's what six pi over five is going to become. Six hundred and sixteen. Hopefully, this made sense, folks. Thanks for watching, and I'll see you in the next one.

Convert the angle $540°$ from degrees to radians.

• A. $3\pi$ rad
• B. $\frac{3\pi}{2}$ rad
• C. $\frac{5\pi}{2}$ rad
• D. $4\pi$ rad

Answer: (A)

Convert the angle $-\frac{5\pi}{6}$ from radians to degrees.

• A. $-120\degree$
• B. $-150\degree$
• C. $-210\degree$
• D. $-240\degree$

Answer: (B)

Here’s what students ask on this topic:

What is the difference between positive and negative angles?

Positive angles are measured in the counterclockwise direction from the positive x-axis, while negative angles are measured in the clockwise direction. This convention helps standardize how angles are represented on the x-y coordinate system. For example, a positive angle of 60° would sweep upward from the x-axis, while a negative angle of -60° would sweep downward. This distinction is crucial when graphing angles or solving problems involving rotations, as the direction determines the terminal side's position. Remember, the initial side always lies along the positive x-axis.

How do you find coterminal angles?

Coterminal angles share the same terminal side but differ by multiples of 360°. To find coterminal angles, simply add or subtract multiples of 360° to the given angle. For example, if the angle is 45°, adding 360° gives 405°, and subtracting 360° gives -315°. Both 405° and -315° are coterminal with 45°. This concept is useful for simplifying angles or working with angles outside the standard range of 0° to 360°.

What is a radian, and how does it relate to degrees?

A radian is a unit of angle measurement based on the circle's radius. One radian is the angle formed when the arc length equals the radius of the circle. A full circle is 2π radians, equivalent to 360°. To convert between radians and degrees, use the formulas: $\theta =\theta \times \frac{\pi }{180}$ for degrees to radians, and $\theta =\theta \times \frac{180}{\pi }$ for radians to degrees. For example, 90° converts to π/2 radians, and π radians converts to 180°.

How do you convert degrees to radians?

To convert degrees to radians, multiply the degree measure by π/180. The formula is: $\theta =\theta \times \frac{\pi }{180}$. For example, to convert 120° to radians: $120\times \frac{\pi }{180}=\frac{2}{3}\pi$. Thus, 120° equals 2π/3 radians. This conversion is essential for working with trigonometric functions and graphing in radians.

What are the classifications of angles based on their measure?

Angles are classified into three categories based on their measure: acute, obtuse, and right angles. Acute angles measure less than 90°, such as 45°. Obtuse angles measure greater than 90° but less than 180°, such as 120°. Right angles measure exactly 90°. These classifications are important for understanding geometric properties and solving problems involving triangles and other shapes.

How do you graph angles in standard position?

To graph angles in standard position, start with the initial side along the positive x-axis. Measure the angle counterclockwise for positive angles and clockwise for negative angles. The terminal side is the line segment that completes the angle. For example, a 60° angle sweeps upward from the x-axis, while a -60° angle sweeps downward. Use the x-y coordinate system as a guide to ensure accuracy when sketching angles.