For more information, please see full course syllabus of Trigonometry
For more information, please see full course syllabus of Trigonometry
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Angles
Main definitions and formulas:
- Degrees are a unit of measurement by which a circle is divided into 360 equal parts, denoted 360^{°} .
- Radians are a unit of measurement by which a circle is divided into 2π parts, denoted 2π^{R}.
- Since the circumference of a circle is 2πr, this means that a 1^{R} angle cuts off an arc whose length is exactly equal to the radius. (It is [1/(2π )] of the whole circle.)
- Since 2π≈ 6.28..., this means that 1^{R} is about one sixth of a circle. But we seldom use whole numbers of radians. Instead we use multiples of π . For example, (π /2)^{R} is exactly one fourth of a circle.
degree measure × π 180= radian measure radian measure × 180 π= degree measure - Coterminal angles are angles that differ from each other by adding or subtracting multiples of 2π^{R} (i.e. 360^{°} ). If you graph them in the coordinate plane starting at the x-axis, they terminate at the same place.
- Complementary angles add to (π /2)^{R}(i.e. 90^{°} ).
- Supplementary angles add to π^{R} (i.e. 180^{°} ).
Example 1:
If a circle is divided into 18 equal angles, how big is each one, in degrees and radians?Example 2:
- Convert 27^{°} into radians.
- Convert (5π /12)^{R} into degrees.
Example 3:
For each of the following angles, determine which quadrant it is in and find a coterminal angle between 0 and 360^{°} or between 0 and 2π^{R}.- 1000^{°}
- − (19π /6)^{R}
- -586^{°}
- (22π /7)^{R}
Example 4:
Convert the following "common values" from degrees to radians: 0, 30, 45, 60, 90. Find the complementary and supplementary angles for each one, in both degrees and radians.Example 5:
For each of the following angles, determine which quadrant it is in and find a coterminal angle between 0 and 360^{°} or between 0 and 2π^{R}.- − (5π /4)^{R}
- 735^{°}
- − (7π /3)^{R}
- -510^{°}
Angles
- In order to calculate the degrees and radians, recall that a circle is 360^{°}
- Calculate the measure of each angle in degrees first
- [360/12]^{°} = 30^{°}
- In radians, we know that 360^{°} is 2π. So, we can calculate the measure of each angle in radians
- [(2π)/12] = [(π)/6]
- Recall the equation for converting degrees to radians
- degree measure ×[(π)/(180^{°})] = radian measure
- 185^{°} ×[(π)/(180^{°})] =
- [(185^{°}π)/(180^{°})]
- The degrees will cancel out, now simplify your answer
- Recall the equation for converting radians to degrees
- radian measure ×[(180^{°})/(π)] = degree measure
- [(7π)/10] ×[(180^{°} )/(π)] =
- 7 ×18^{°}
- radian measure ×[(180^{°})/(π)]= degree measure
- [(3π)/5] ×[(180^{°})/(π)] =
- [(540^{°}π)/(5π)]
- The radians cancel so now just divide
- Recall the formula for converting degrees to radians
- degree measure ×[(π)/(180^{°})] = radian measure
- 76^{°}× [(π)/(180^{°} )] =
- [(76^{°} π)/(180^{°} )]
- The degrees will cancel. Simplify your fraction
460^{°}
- Notice that 460^{°} is larger than 360^{°}, so we must subtract 360^{°} from our given angle until we reach an angle that is between 0^{°} and 360^{°}
- 460^{°} - 360^{°} = 100^{°} which is an angle that is between 0^{°} and 360^{°} and it is coterminal to 450^{°}
- Now we can determine which quadrant our angle is in by using the following:
Quadrant I has angles between 0^{°} and 90^{°}
Quadrant II has angles between 90^{°} and 180^{°}
Quadrant III has angles between 180^{°} and 270^{°}
Quadrant IV has angles between 270^{°} and 360^{°}
[( − 13π)/15]
- Notice that [( − 13π)/15] is smaller than 0^{°}, so we must add 360^{°} or 2π to our given angle until we reach an angle that is between 0 and 360^{°} (i.e. between 0 and 2π)
- [( − 13π)/15] + 2π = [(17π)/15] which is an angle that is between 0 and 2π and it is coterminal to [( − 13π)/15]
- Now we can determine which quadrant our angle is in by using the following:
Quadrant I has angles between 0 and [(π)/2]
Quadrant II has angles between [(π)/2] and π
Quadrant III has angles between π and [(3π)/2]
Quadrant IV has angles between [(3π)/2]and 2π
a. 47^{°}
b. [(π)/12]
- Recall that complementary angles add to 90^{°} or [(π)/2]
b. [(π)/2] − [(π)/12] = [(5π)/12]
a. 114^{°}
b. [(4π)/5]
- Recall that supplementary angles add to 180^{°} or π
- a. 180^{°} − 114^{°} = 66^{°}
b. π− [(4π)/5] = [(π)/5]
b. [(π)/5]
a. [( − 6π)/11]
b. 623^{°}
c. [( − 27π)/13]
d. 1572^{°}
- a. [( − 6π)/11]is smaller than 0 so you have to add 2π to find a coterminal angle
- [( − 6π)/11] + 2π = [(16π)/11] which is in quadrant III
- b. 623^{°} is larger than 360^{°} so you have to subtract 360^{°} to find a coterminal angle
- 623^{°} − 360^{°} = 263^{°} which is in quadrant III
- c. [( − 27π)/13] is smaller than 0 so you have to add 2p to find a coterminal angle
- [( − 27π)/13] + 2π = [( − π)/13] which is still smaller than 0 so keep adding 2p
- [( − π)/13] + 2π = [(25π)/13] which is in quadrant IV
- d. 1572^{°} is larger than 360^{°} so you have to subtract 360^{°} to find the coterminal angle
- 1572^{°} − 360^{°} = 1212^{°} which is still larger than 360^{°} so keep subtracting by 360^{°}
- 1212^{°} − 360^{°} = 852^{°}
852^{°} − 360^{°} = 492^{°}
492^{°} − 360^{°} = 132^{°} which is in quadrant II
b. 263^{°}; III
c. [(25π)/13]; IV
d. 132^{°}; II
*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Answer
Angles
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
- Intro
- Degrees
- Radians
- Converting Between Degrees and Radians
- Coterminal, Complementary, Supplementary Angles
- Example 1: Dividing a Circle
- Example 2: Converting Between Degrees and Radians
- Example 3: Quadrants and Coterminal Angles
- Extra Example 1: Common Angle Conversions
- Extra Example 2: Quadrants and Coterminal Angles
- Intro 0:00
- Degrees 0:22
- Circle is 360 Degrees
- Splitting a Circle
- Radians 2:08
- Circle is 2 Pi Radians
- One Radian
- Half-Circle and Right Angle
- Converting Between Degrees and Radians 6:24
- Formulas for Degrees and Radians
- Coterminal, Complementary, Supplementary Angles 7:23
- Coterminal Angles
- Complementary Angles
- Supplementary Angles
- Example 1: Dividing a Circle 10:38
- Example 2: Converting Between Degrees and Radians 11:56
- Example 3: Quadrants and Coterminal Angles 14:18
- Extra Example 1: Common Angle Conversions
- Extra Example 2: Quadrants and Coterminal Angles
Trigonometry Online Course
I. Trigonometric Functions | ||
---|---|---|
Angles | 39:05 | |
Sine and Cosine Functions | 43:16 | |
Sine and Cosine Values of Special Angles | 33:05 | |
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D | 52:03 | |
Tangent and Cotangent Functions | 36:04 | |
Secant and Cosecant Functions | 27:18 | |
Inverse Trigonometric Functions | 32:58 | |
Computations of Inverse Trigonometric Functions | 31:08 | |
II. Trigonometric Identities | ||
Pythagorean Identity | 19:11 | |
Identity Tan(squared)x+1=Sec(squared)x | 23:16 | |
Addition and Subtraction Formulas | 52:52 | |
Double Angle Formulas | 29:05 | |
Half-Angle Formulas | 43:55 | |
III. Applications of Trigonometry | ||
Trigonometry in Right Angles | 25:43 | |
Law of Sines | 56:40 | |
Law of Cosines | 49:05 | |
Finding the Area of a Triangle | 27:37 | |
Word Problems and Applications of Trigonometry | 34:25 | |
Vectors | 46:42 | |
IV. Complex Numbers and Polar Coordinates | ||
Polar Coordinates | 1:07:35 | |
Complex Numbers | 35:59 | |
Polar Form of Complex Numbers | 40:43 | |
DeMoivre's Theorem | 57:37 |
Transcription: 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
1 answer
Last reply by: Dr. William Murray
Fri Nov 3, 2017 6:37 PM
Post by Julius Francis on November 1, 2017
Do you recommend any Trigonometry Workbooks?
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Last reply by: Dr. William Murray
Tue Jul 18, 2017 8:32 AM
Post by Mohamed E Sowaileh on July 10, 2017
Hello Dr. William Murray,
I hope you are very well.
I am a student who is extremely weak in math. In order to be very strong in math, specially for engineering field, could you provide me with sequential order of mathematical topics and textbooks. With what should I begin so that I can master big topics like calculus, statistics, probability ... etc.
Your guidance is precious to me.
Thank you so much.
1 answer
Last reply by: Dr. William Murray
Sat Mar 18, 2017 11:56 PM
Post by Anna Kopituk on March 15, 2017
I am taking this as a home school student. How can I be tested on this material and come up with a grade?
1 answer
Last reply by: Dr. William Murray
Mon Jan 9, 2017 10:20 AM
Post by Anish Srinivasan on January 8, 2017
Is there a complement and supplement of a negative angle, and can there be a complement angle of an angle greater than 90 degrees? Also, can there be a supplement angle of an angle greater than 180 degrees?
3 answers
Last reply by: Dr. William Murray
Mon Nov 7, 2016 11:39 AM
Post by Firebird wang on November 2, 2016
Professor, I know that AP Statistics is not your subject, but I just wonder if you are able to watch the two videos which called Practice Test 2013 AP Statistics an Practice Test 2014 AP Statistics in the AP Statistics content? Both videos showing network error, I dont know why. I already tried in different computers already.
2 answers
Last reply by: Dr. William Murray
Mon Jun 13, 2016 8:53 PM
Post by Tiffany Warner on June 10, 2016
Hello Dr. Murray,
I see lots of comments regarding practice problems and such, but I see no link for them like I did with other lectures. Your examples in the video were definitely helpful for making the concepts sink in, but I do like to do practice problems every once in awhile to test myself and see if I really got it. Did they take them away?
Thank you!
1 answer
Last reply by: Dr. William Murray
Fri Oct 30, 2015 4:23 PM
Post by Alexander Roland on October 30, 2015
Hello Professor,
If you don't mind sharing, what type of technology is that you are using to deliver instructions?
Thanks for sharing
4 answers
Last reply by: Dr. William Murray
Wed Jun 17, 2015 10:18 AM
Post by Ashley Haden on April 29, 2015
It seems a bit confusing that radians are 2PIR. Why isn't there a symbol for a radian, or just for 2PI?
1 answer
Last reply by: Dr. William Murray
Mon Aug 4, 2014 7:31 PM
Post by Tehreem Lughmani on July 9, 2014
Example 3 how to know what is between 0-2pi? I'm not good with fractions :)
1 answer
Last reply by: Dr. William Murray
Mon Aug 4, 2014 7:13 PM
Post by Tehreem Lughmani on July 9, 2014
Example 3 - C. -586+360= -226+360= 134???
-226+360= 94
I get this answer every single time~ what's wrong here -.-|||
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Last reply by: Dr. William Murray
Wed Oct 9, 2013 5:48 PM
Post by Rakshit Joshi on October 7, 2013
How to download the notes??
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Wed Oct 9, 2013 5:47 PM
Post by Rakshit Joshi on October 6, 2013
Sir you are AWESOME!!
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Wed Aug 14, 2013 12:53 PM
Post by Reema Batra on August 1, 2013
I found errors in questions 6 and 7 in the practice problems...
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Last reply by: Dr. William Murray
Wed Aug 14, 2013 12:53 PM
Post by Reema Batra on August 1, 2013
For the sixth practice problem, I found an error:
Question - Determine which quadrant the following angle is in and find a coterminal angle between 0 and 360: 450.
My Answer: 90; y-axis.
The Answer Given: 115; Quadrant 2. This also had 450-360 is 115. This is incorrect, if I am not mistaken...
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Fri Jul 5, 2013 9:57 AM
Post by mohammad sawari on July 5, 2013
what is call the half of the radius
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Fri Jul 5, 2013 9:55 AM
Post by Norman Cervantes on July 1, 2013
second time going through this course. going straight to the examples, this course is very well taught. it really gives my brain a workout!
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Last reply by: Dr. William Murray
Mon Jun 10, 2013 7:34 PM
Post by Dr. Son's Statistics Class on June 10, 2013
You're a great professor!
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Last reply by: Dr. William Murray
Mon Jun 10, 2013 7:33 PM
Post by Jorge Sardinas on June 8, 2013
i am 9
3 answers
Last reply by: Dr. William Murray
Sat Jun 8, 2013 5:44 PM
Post by Manfred Berger on May 29, 2013
Could you elaborate a bit on what the motivation behind using signed angles is? Quite frankly I fail to see a functional difference between an angle -x and x+180 degrees.
3 answers
Last reply by: Dr. William Murray
Wed May 29, 2013 11:17 AM
Post by Manfred Berger on May 28, 2013
Your Rs look a lot like exponents. Is that a general notation or just your handwriting?
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Thu Apr 25, 2013 3:04 PM
Post by Edmund Mercado on April 15, 2012
A very fine presentation.
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Thu Apr 25, 2013 3:02 PM
Post by Levi Stafford on March 19, 2012
commenting on the text on the quick notes.
"2Ï€ parts, denoted 2Ï€ R." the pi's look like "n's" and it is confusing...I thought they were variables.
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Last reply by: Dr. William Murray
Thu Apr 25, 2013 2:59 PM
Post by kirill frusin on March 2, 2012
I believe you confused compliment and supplement in one of your videos. The video I watched before this about RADIANS says supplement is two angles added to be 90 degrees and complimentary add to 180 degrees.
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Thu Apr 25, 2013 2:53 PM
Post by Janet Wyatt on February 10, 2012
Is there practice worksheets I can print?
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Thu Apr 25, 2013 2:50 PM
Post by Valtio Cooper on January 14, 2012
Great lecture! I got it but I'm having a problem with a question that I got for homework pertaining to this topic! I was wondering if i could be given some guidelines if possible please.
The question is:
A Hexagon is inscribed in a circle. if the difference between the area of the circle and the area of the hexagon is 24meters squared use the formula for the area of the sector to approximate the radius of the circle.
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Thu Apr 25, 2013 2:18 PM
Post by Kyle Spicer on December 6, 2011
where do you take the assessment test? I can't find it.
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Thu Apr 25, 2013 2:16 PM
Post by Robert Reynolds on October 22, 2011
Thumbs up for 2 things:
1. Assessment test at the beginning to find where you are at now.
2. End of lesson tests.
That make this site the Deathstar of education. (Without the silly hole that you can shoot down and blow the whole thing.)
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Last reply by: Dr. William Murray
Thu Apr 25, 2013 2:15 PM
Post by David Burns on August 8, 2011
I wish this site had tests available, or at least links to them. Other than that I love it here.
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Last reply by: Dr. William Murray
Thu Apr 25, 2013 2:12 PM
Post by Sheila Greenfield on March 3, 2011
i get this and i'm a freshman in high school i really like this cant wait to learn more
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Thu Apr 25, 2013 2:10 PM
Post by Erin Murphy on March 16, 2010
You are a fantastic prof. On to my next lesson!