Trigonometry Angles

Hi we are here to do some extra examples on measuring angles and converting back and forth between degrees and radians.0000

I hope you had a chance to try this out on your own a little bit.0008

There are common values that you use a lot in all kinds of trigonometric functions and situations.0012

It is worth at least working them out once on your own and memorize them after that.0020

The common values are 0 degrees, 30 degrees, 45degrees, 60degrees, and 90 degrees.0027

What we are going to do is find the complementary and supplementary angles for each one in both degrees and radians.0037

The reason these angles are so important is because 90 degrees is a right angle0047

What we are doing is chopping a right angle up into either two equal pieces which gives us 45 degrees or three equal pieces which gives us the 30 degrees and 60 degrees angles.0054

Those are very common ones that come up very often.0065

It is worth knowing what these are in both degrees and radians.0068

Knowing their complements and supplements and knowing what the complements and supplements are in degrees and radians as well.0073

Let me make a little chart here, we are starting out in degrees.0082

We have the 0 degrees angle, 30 degrees, 45 degrees, 60 degrees, and 90 degrees.0087

Let me convert those into radians first.0098

The 0 degrees angle is still 0 in radians.0102

Well remember that a 90 degrees angle is pi over 2 radians.0107

A 30 degrees angle is 1/3 of that, it’s pi over 2 divided by 3, that is pi over 6.0114

45 degrees is 90 degrees divided by 2, that is pi over 2 divided by 2 which is pi over4.0123

60 degrees is twice 30 degrees, 60 degrees is 2 x pi over 6, 2 pi over 6 is pi over 3.0131

Those are pretty convenient fraction if you write them in radians.0143

The complementary angles we will do it in terms of degrees first.0147

Remember complementary angles add up to 90 degrees.0154

If you know what the angle is, you will do 90 degrees minus that number to get the complementary angle.0158

If you start with the 0 degrees angle, the complementary angle is 90 degrees because 90 minus 0 is 90.0165

30 degrees angle the complementary angle is 60 because those add up to 90.0173

45 degrees angle is its own complement because 45 and 45 is 90.0178

60 degrees angle its complement is 30.0185

A 90 angle its complement is 0.0187

Let us do those in terms of radiance0192

The 0 radiant angle its complement is going to be pi over 2 because those add up to pi over 2.0198

Remember that is the same as a 90 degree angle.0207

Pi over 6 + pi over 3 is pi over 2.0210

You can work out the fractions there or you can remember that 30 + 60 equals 90.0216

45 degree angle or pi over 4 is its own complement.0222

Pi over 3 we already figured out that its complement is pi over 6.0228

Pi over 2 its complement is 0 because those add up to pi over 2 or 90 degree angle.0235

Let us figure out the supplementary angles.0243

Remember supplementary angle means that they add up to 180 degrees.0246

In each case we are looking for what adds up to pi over 2 or 90.0251

We start with 0, that means the supplement is 180 itself.0256

If we start with 30 it is 150.0262

45 a 180 - 45 is 135.0265

180 - 60 is 120.0271

180-90 is 90 itself.0275

Finally, if we find the supplementary angles in terms of radians.0280

Remember we are looking here for, before they add it up to a 180 degrees in terms of radians they should add it up to pi.0286

0 + pi adds up to pi.0295

Pi over 6 + 5 pi over 6 adds up to pi.0299

Pi over 4 + 3 pi over 4 adds up to pi.0306

Pi over 3 + 2 pi over 3 adds up to pi.0313

Pi over 2 + pi over 2 adds up to pi.0320

All of these are common values.0326

These are all conversions back and forth between degrees and radians that you should now very well as a trigonometry student just because these angles come up so often.0328

It is probably worth understanding the pictures behind each of these numbers that I have written down.0337

For example, when we look at complementary angles.0344

Here is a 30 degree angle and its complement is a 60 degree angle.0349

In terms of radians, that is pi over 6 radians.0355

The 60 degree angle is pi over 3 radians.0362

Add those together, you will get pi over 3 + pi over 6 adds up to pi over 2.0367

Here is another right angle and I’m dividing it in 2 into 45 degrees, which is pi over 4 radians.0377

We see that that angle is its own complement 45 degrees is pi over 4 radians.0385

I will do the supplements in blue.0394

If we start out with a 30 degree angle or pi over 6 radians.0399

Then its supplement is a 150 degree angle which is 5 pi over 6.0411

If we start out with a 60 degree angle which is pi over 3 radians then its supplement.0423

Remember to put them together and they are supposed to make 180 degree or pi radians is 120 degrees which is 2 pi over 3 radians.0436

Finally if you start out with a 45 degree angle which is pi over 4 radians then its supplement is 135 degrees which is 3 pi over 4 radians.0452

All of those are angles that you should know very well both in terms of degrees and radians because we will be seeing a lot of them in our trigonometry lessons.0475

Finally, our example here is for each of the following angles.0000

I want to find out what quadrant it is in.0005

I want to find the coterminal angle between 0 and 360 degrees over 0 and 2 pi radians.0009

We got 4 angles here and I’m going to give you in both degrees and radians.0018

We are going to start out with -5 pi over 4 radians and that will do 735 degrees, -7 pi/3 radians and -510 degrees.0027

If you want you can try those on your own.0047

I will help you out starting with -5 pi/4 radians.0051

Ok, that is not between 0 and 2 pi radians.0057

What we are going to is add a 2 pi to it, 2 pi is 8 pi/4, (8 pi/4 – 5 pi/4) is 3 pi/4.0060

Let us graph that, 0 pi/2 pi, 3 pi/2, and 2 pi, 3 pi/4 is between pi/2 and pi.0076

It is right there, that is in the second quadrant.0094

Our answer here is 3 pi/4 and it is in the second quadrant.0107

The way you can understand how that came from the original -5 pi/4.0115

If you went in the other direction 5 pi/4 in the other direction from the traditional direction because it is negative. You will end up at that angle.0121

Its coterminal angle is 3 pi over 4 in quadrant 2.0131

735 degrees that is way bigger than 360.0136

Let us drop down multiples of 360, we can actually take out two multiples of 360 right away.0141

I’m going to subtract 720 degrees and we get 15 degrees.0148

If you write things in terms of degrees here, 0 is 0 degrees, Pi/2 is 90 degrees, Pi is 180 degrees, 3 pi/2 is 270 degrees, 2pi is 360 degrees.0154

15 degrees is between 0 and 90. That is right about there0174

What that means is 735 would actually go out in circle twice and end up right at the same place that 15 degrees ended up.0182

So, the coterminal angle is 15 degrees and that is in the first quadrant.0192

Let us look at -7pi/ 3, that is less than 0 so I’m going to add 2 pi to that, plus 2 pi, well 2 pi is 6 pi/3.0203

That would give us – pi/3, that is still less than 0, let me add another 2 pi.0217

That is again 6 pi/3 – pi/3, would give us 5 pi/3, that is between 0 and 2 pi.0227

Now, we just have to figure what out quadrant it is in.0238

5 pi/3 is a little bit bigger than 3 pi/2, 3 pi/2 is 1 ½, 5 pi/3 is 1.67.0241

So, 5 pi/3 is a little bit bigger than 3 pi/2.0252

Let me erase some of these extra angles, they are getting in our way.0260

I will show you where 5 pi over 3 is.0273

If you go between pi and 2 pi, there is 4 pi/3, there is 5 pi/3.0277

And so, 5 pi/3 is right there, that is clearly in the fourth quadrant.0283

Finally, -510 degrees, where does that one end up? That is definitely less than 0.0295

So, I will add 360 degrees and we will end up with -150 degrees, still less than 0 there so I will add another 360 degrees, that gives us 210 degrees.0303

That is between 0 and 360 degrees, we know we found our coterminal angle, 210 degrees is just past 180 degrees.0322

In fact, it is 30 degrees past 180 degrees, it is right there, that is in the third quadrant.0333

Just to recap here, finding these coterminal angles and finding out what quadrant they are in.0346

To find the coterminal angle, you take the angle that you are given in degrees or radians and you add or subtract multiples of either 2 pi radians or 360 degrees.0351

Then you add or subtract these multiples until you get them in to the range that you want, 0 to 2 pi radians, 0 to 360 degrees.0357

Once you get it into those ranges then you can break it down finally and ask are you between 0, pi over 2, pi, 3 pi over 2, or 2 pi.0380

That tells you which quadrant you are in or in terms of degrees that would be 0, 90 degrees, 180 degrees, 270 degrees, or 360 degrees.0394

That will tell you what quadrant you are in if you are in terms of degrees.0407

That is the first or our lessons on trigonometry for www.educator.com.0412

Here we talked about angles, we have not really got in to the trigonometric functions yet.0417

In the next lessons we will start talking about sine and cosine, all the different identities and relationships,how you use them in triangles?0421

We will talk about tangents and secants.0429

That is all coming in the later lectures on www.educator.com.0431

Hi, This is Will Murray and I'm going to be giving the trigonometry lectures for educator.com.0000

We're very excited about the trigonometry series.0006

In particular, for me, trigonometry is the class that got me excited about math.0008

I'm really looking forward to working with you on learning some trigonometry.0013

We're going to start right away here, learning about angles.0018

The first thing you have to understand is that there's two different ways to measure angles.0021

People use degrees which you probably already heard of, and radians which you may not hear about until you start to take your first trigonometry class.0029

They're just two different ways in measuring.0037

You can use either one but you really need to know how to use both, and convert back and forth.0041

That's what I'll be covering in this first lecture.0044

We'll start with degrees.0048

Degrees are unit of measurement in which a circle gets divided into 360 degrees0050

If you have a full circle, the whole thing is 360 degrees.0059

That's 360 degrees.0066

Then if you have just a piece of a circle, then it gets broken up into smaller chunks.0067

For example, an angle that's half of a circle here that's 180 degrees, because that's half of 360, a quarter of a circle which is a right angle, that would be 90 degrees, then so on.0072

You can take a 90-degree angle and break it up into two equal pieces.0093

Then each one of those pieces would be a 45-degree angle.0098

Or you could break up a 90-degree angle into three equal pieces, and each one of those would be a 30-degree angle.0104

We'll be studying trigonometric functions of these different angles.0114

In the meantime, it's important just to get comfortable with measuring angles in terms of degrees.0118

The second unit of measurement we're going to use to measure angles is called radians.0124

That's a little bit more complicated.0129

You probably haven't learned about this until you start to study trigonometry.0131

The idea is that, you take a circle, and remember that the circumference of a circle is equal to 2Π times the radius, that's 2Π r, it's one of those formulas that you learned in geometry.0135

What you do with radians is, you break the circle up, and you say the entire circle is 2Π radians.0151

2Π radians.0159

What that means is that a one-radian angle, well, if the entire circle is 2Π radians, then 1 radian, use a little r to specify the radians, cuts off an arc that is 1 over 2Π of the whole circle.0161

One radian, an angle that is 1 radian cuts of a fraction of the circle that is 1 over 2Π.0184

Since the radius is 2Π times r, sorry, the circumference is 2Π times r, if you have 1 over 2Π of the whole circumference, what you get is exactly the length of the radius.0194

That's why they're called radians is because if you take one-radian angle, it cuts of an arc length that is exactly equal to the radius.0214

That's the definition of radians.0226

It takes a little bit of getting used to.0228

What you have to remember that's important is that the whole circle is 2Π radians.0230

That means a half circle, a 180-degree angle, is Π radians.0240

A right angle, a 90-degree angle, or a quarter circle is Π over 2 radians, and so on.0250

Then you can break that down into the even smaller angles like we talked about before.0262

If you take a right angle and you cut it in half, so that was a 45-degree angle before, that's Π over 4 radians because it's half of Π over 2.0268

If you take a right angle and you cut it into three equal pieces, so those are 30-degree angles before, in terms of radians, that's Π over 2 divided by 3, so that's Π over 6 radians.0281

You want to practice going back and fort between degrees and radians, and kind of getting and into the feel of how big angles are in terms of degrees and radians.0299

We'll practice some of that here in this lecture.0307

Remember that Π is about 3.14, so 2Π is about 6.28.0310

That means we're breaking an entire circle up in the 2Π radians, so the circle gets broken up into about 6.28 radians.0317

That means one radian is about one-sixth of a circle.0326

What I've shown up here is pretty accurate that one radian is about one-sixth of a circle.0332

It's about 60 degrees but it's not exact there because it's not exactly 6 it's 6.28 something.0339

That's about roughly a radian is, about 60 degrees, but we really don't usually talked about whole numbers or radians.0347

People almost always talk about radians in multiples of Π the same way I was doing here, where I said the circle is 2Π radians, the half circle is Π radians, the right angle is Π over 2.0355

People almost always talk about radians in multiples of Π and degrees in terms of whole numbers.0366

Sometimes, they don't even bother to write the little r.0374

It's just understood that if you're using a multiple of Π , then you're probably talking about radians.0376

Let's practice going back and forth between degrees and radians.0385

Remember that 360 degrees is a whole circle.0389

That's 2Π radians.0394

What that means is that Π radians is equal to 360 degrees over 2, which is 180 degrees.0398

Pi radians is 180 degrees.0412

That gives you the formula to convert back and forth between degree measurement and radian measurement.0415

If you know the measurement in degrees, you multiply by Π over 180 and that tells you the measurement in radians.0421

If you know the measurement in radians, you just multiply by 180 over Π , and that tells you the measurement in degrees.0429

We'll practice that in some of the examples later on.0439

We got a few more definitions here.0442

Coterminal angles, what that means is that their angles that differ from each other by a multiple of 2Π radians,0446

remember that's a whole circle, or if you think about it in degrees, 360 degrees.0456

For example, if you take a 45-degree angle, and then you add on 360 degrees, that would count as a 360 plus 45 is 405 degrees.0462

Forty-five and 405 degrees are coterminal.0484

In the language of radians, 45 degrees is Π over 4.0491

If you add on 2Π radians, if you add on a whole circle to that, you would get...0497

This should end up here.0511

If you add on 2Π plus Π over 4, well 2Π is 8Π over 4, so you get 9Π over 4.0512

Then Π over 4 and 9Π over 4 are coterminal angles.0522

The reason they're called coterminal angles is because we often draw angles starting with one side on the positive x-axis.0528

We start with one side on the positive x-axis.0539

I'll draw this in blue.0544

Then we draw the other side of the angle just wherever it ends up.0545

Coterminal angles are angles that will end up at the same place, that's why they're called coterminal.0550

If they differ from each other, if one is 2Π more than the other one, or 360 degrees more than the other one, or maybe 720 degrees more than the other one, then we call them coterminal because they really end up on the same terminal line here.0561

Couple other definitions we need to learn.0580

Complementary angles are angles that add up to being a right angle, in other words, 90 degrees or Π over 2.0583

If you have two angles, like this two angles right here , that add up to being a right angle, 90 degrees or Π over 2, those are complementary.0591

Supplementary angles are angles that add up to being a straight line, in other words, Π radians or 180 degrees.0608

Those two angles right there are supplementary.0620

That's all the vocabulary that you need to learn about angles, but we'll go through and we'll do some examples of each one to give you some practice.0629

Here's our first example.0640

If a circle is divided into 18 equal angles, how big is each one in degrees and radians?0642

Let me try drawing this.0648

We've got this circle and it's divided into a whole bunch of little angles but each one is the same.0653

We want to figure out how big each one is, in terms of degrees and radians.0664

Let's solve this in degrees first.0668

Remember that a circle is 360 degrees.0671

If it's divided into 18 parts, then each part will be 20 degrees.0676

Each one of those angles will be 20 degrees.0682

We've done the degree one, how about radians?0686

Remember that an entire circle is 2Π radians.0691

If we divide that by 18, then we get Π over 9 radians will be the size of each one of those little angles.0695

You can measure this angle either way, we say 20 degrees is equal to Π over 9 radians.0706

Second example here, we want to convert back and forth between degrees and radians.0718

Let's practice that.0721

We want to convert 27 degrees into radians.0724

Well, let's remember the formula here, the conversion formula, is Π over 180.0727

So we do 27 times Π over 180.0733

That's our conversion formula from degrees into radians.0738

I'm just going to leave the Π because it doesn't really cancelled anything.0743

The 27 over 180 does simplify.0747

I could take a 9 out of each ones.0752

That would be 3.0753

If we take a 9 out of 180, then there'd be 20.0755

What we end up with is 3Π over 20 radians, as our answer there.0761

Converting back and forth between degrees and radians is just a matter of remembering this conversion factor, Π over 180 gets you from degrees into radians.0777

For the second part of this example, we're given a radian angle measurement, 5Π over 12.0789

We want to convert that into degrees, 5Π over 12 radians.0794

We just multiply by the opposite conversion factor, 180 over Π.0804

Let's see here.0813

The pis cancel.0814

One hundred eighty over 12 is 15.0818

That's 5 times 15 degrees.0826

That gives us 75 degrees.0833

The same angle that you would measure, in radians is being 5Π over 12, will come out to be a 75-degree angle.0838

Converting back and forth there is just a matter of remembering the Π and the 180, and multiplying by one over the other to convert back and forth.0848

Third example is some practice with coterminal angles.0859

In each case, what we want to do is, we're given an angle and we want to find out what quadrant it's in.0863

That's assuming that all the angles are drawn in the standard position with their starting side on the positive x-axis.0872

We want to start on the positive x-axis.0882

We want to see which one of the four quadrants the angle ends up in.0885

Then we want to try to simplify these angles down by finding a coterminal angle that's between 0 and 360, or between 0 and 2Π radians.0893

Let's start out with 1000 degrees.0904

A thousand degrees is going to be, that's way bigger than 360.0908

Let me just start subtracting multiples of 360 from that.0912

If I take off 360 degrees, what I'm left with is 640 degrees.0916

That's still way bigger than 360 degrees.0926

I'll subtract off another 360 degrees and what I'm left with is 280 degrees.0928

That's between 0 and 360.0940

I found my coterminal angle there.0942

I wanna figure which quadrant it's gonna end up in.0945

Now, remember, if we start with 0 degrees being on the x-axis, that would make 90 degrees being on the positive y-axis.0947

Then over here on the negative x-axis, we'd have 180 degrees.0959

Down here is 270 degrees, because that's 180 plus 90.0964

Then 360 degrees would be back here at 0 degrees.0971

Two hundred and eighty degrees would be just past 270 degrees.0976

That's a little bit bigger than 270 degrees.0981

It would be about right there.0983

That's 280.0986

That puts it in the fourth quadrant.0990

Next one's a radian problem.1001

We have -19Π over 6 radians.1005

That's one, I'll do this one red.1011

That's one that goes in the negative direction.1014

We start on the positive x-axis but now we go in the negative direction.1016

Instead of going up around past the positive y-axis, we go down in the negative direction and we go -19Π over 6.1021

If you think about it, 19Π over 6 is bigger than 2Π.1031

Let me start with 19Π over 6 and subtract off a 2Π there.1039

Well, 2Π is 12Π over 6, so that gives us 7Π over 6.1044

If we do, -19Π over 6 plus 2Π, that will give us, -19Π over 6 plus 12Π over 6 is -7Π over 6.1055

That's still not in the range that we want, because we want it to be between 0 and 2Π radians.1070

Let me add on another 2Π plus 2Π gives us positive 5Π over 6.1076

The trick with finding these coterminal angles with degrees, it was just a matter of adding or subtracting 360 degrees at a time.1084

With radians, it's a matter of adding or subtracting 2Π radians at a time.1092

Remember, 2Π is a whole circle.1098

We end up with 5Π over 6.1101

That is between 0 and 2Π so we're done with that part but we still have to figure out what quadrant it's in.1104

Well now 5Π over 6, where would that be?1111

Well if we map out our quadrants here, 0 is right there on the positive x-axis just as we had before, 90 degrees is Π over 2 radians, 180 degrees is Π radians, and 270 degrees is 3Π over 2 radians.1114

Then 360 degrees is 2Π radians, a full circle.1135

Where does 5Π over 6 land?1143

Well that's bigger than Π over 2, it's less than Π, so, 5Π over 6 lands about right there.1145

That's in the second quadrant.1155

OK, we have another degree one.1165

Negative 586 degrees, and what are we going to do with that?1168

It's going in the negative direction so it's going down south from the x-axis.1173

Negative 586 degrees.1180

Well, 586 is way outside our range of 0 and 360.1183

Let's try adding 360 degrees to that.1187

That gives you -226 degrees, which is still outside of our range.1193

Let's add another 360 degrees.1203

We're adding and subtracting multiples of a full circle 360 degrees.1208

That gives us positive 134 degrees.1212

Now, positive 134 degrees, that is in our allowed range between 0 and 360.1218

So we finished that part of the problem.1227

Where would that land in terms of quadrants?1229

Let me redraw my axis because those are getting a little messy.1230

That's 0, 90, 180, 270 and 360.1237

Where's 134 going to be?1246

A hundred and thirty-four is going to be between 90 and 180, almost exactly halfway between.1250

It's about right there.1255

That's in the second quadrant.1258

The answer to that one is that that's in the quadrant number two there.1262

Finally, we have 22Π over 7, again given in radians.1272

The question is, is that between 0 and 2Π?1281

It's not, it's too big.1284

It's bigger than 2Π.1285

Let me subtract off a multiple of 2Π.1287

I'll subtract off just 2Π, which is 14Π over 7.1291

That simplifies down to 22Π over 7 minus 14Π over 7, is 8Π over 7.1296

Now, 8Π over 7 is between 0 and 2Π.1303

We found our coterminal angle.1309

Where will that land on the axis?1311

Well, remember 0 degrees is 0 radians, 90 degrees is Π over 2 radians, 180 degrees is Π radians, and 270 degrees is 3Π over 2 radians, and finally, 360 degrees is 2Π radians.1312

Eight pi over 7 is just a little bit bigger than 1.1334

That's a little bit, or 8 over 7 is a little bigger than 1.1336

Eight pi over 7 is just a little bit bigger than Π.1341

Let's going to put it about right there which will put it in the third quadrant.1346

Let's recap how we found this coterminal angles.1360

Basically, you're given some angle and you check first whether it's in the correct range, whether it's in between 0 and 2Π radians,1363

or if it's given in degrees, whether it's between 0 and 360 degrees.1372

If it's not already in the correct range, if it's negative or if it's too big, then what you do is you add and subtract multiples of 360 degrees or 2Π radians until you'll get it into the correct range,1378

the range between 0 and 360 degrees or 0 and 2Π radians.1393

Once you get it in that range, if you want to figure out what quadrant it's in, well in degrees, it's a matter of checking 0, 90, 180, 270, and 360;1402

in radians,it's a matter of checking 0, Π over 2, Π, 3Π over 2, 2Π.1417

Which one of those ranges does it fall into?1426

That tells you what quadrant it's in.1428