ALRIGHTY!!!
We are going to use this fact that we just learned which is pi radians equals 180 degrees
to learn how to convert back and forth between degree measures and radian measures.
We are going to start off by taking 40 degrees.
I want to take that 40 degrees and get a measurement that is going to be in radians.
Now by the way, let's throw this theta up here for a minute.
In any kind of advanced math book if you have a Greek symbol that does stand for an angle
measure.
So we have an angle that equals 40 degrees.
I want that in radians.
Well what we want to do is not change the value of rotation, I just want to make it
look different.
So I want to multiply 40 by an equivalent value...
I am going to multiply it by one.
Think of reducing fractions.
If you have ever used a fraction of 2/3, I can multiply the top and bottom by 3 and simply
make it look different.
That 6/9 is still 2/3.
I want to multiply 40 by a value that equivalent to one.
Now numerically it is not going to be one, but geometrically it will be an equivalent
of one.
I want to take 40 degrees and multiply, actually what I want to do, is I want to multiply by
something that is going to divide out the measurement of degrees.
That is going to be 180 degrees.
Then multiply in its radian equivalent.
Well what is equal to 180 degrees as far as Geometry is concerned?
Well, that is as we just learned pi radians.
Now I know pi divided by 180 is not equal to 1.
But in Geometry terms pi gets you half way around the circle, and so does 180 degrees.
So in Geometry terms this is effectively one.
So this becomes 40pi/180.
That is...
You don't have to take this many steps to reduce, but that will be 4pi/18 which is 2pi/9
radians.
Now I do not need to write this word radians to make sure that you know that we are in
radians.
You will need to be very careful in your reading of your advanced math book to identify that
when you see this little circle in the upper right hand corner... that is indicating degrees.
If you do not, that indicates radians.
So me writing this is kind of redundant.
Please don't get sucked into the idea that every time you see pi you are in radians either.
You need to look for that degree mark.
If it is there, you have degrees.
If it is not there, even if it is a decimal, you are in radians.
So if you multiply by pi/180, that is going to get you from degrees to radians.
If you want to go in reverse...
If you want to take an angle measure that is equal to pi/4 radians, there is no degree
mark, and convert that into degrees you will want to multiply by an equivalent of 1 that
will take out the radians and put in it's place 180 degrees.
Now I know again that is not equal to one mathematically or numerically, but in terms
of Geometry this is half way around the circle...
half way around the circle so in Geometry it is effectively one.
The pi is going to cancel out and you get 180 over 4.
This comes out to be 45 degrees, right.
Wait a minute, I just wrote 45 radians.
No, no, no.
45 degrees.
Ok.
So uh...
There you go:)
Ok.
That is converting to degrees to radians and radians to degrees.
Last concept, it is not really a concept but, we are also going to try and build up to the
idea of a unit circle.
Any work that you do in Trig, you teacher should be requiring you to have all of those
answers in exact form if you are using an angle on something called a unit circle.
Now, I am just going to show you how to come up with the angles around the unit circle.
I am not going to develop the whole concept right now.
But let's build up and show you the 16 angles that you have to memorize for the concept
of the unit circle which is coming up very very soon.
Ok, that is a little better.
Ok, a rough idea of a circle.
We are going to have zero degrees.
Now 16 angles are around the circle.
You are going to find those 16 angles by counting by 30 degrees.
We are going to degrees first and then go into radians.
I am going to put my corners in, if you will, or my 90 degree marks so I get some kind correct
spacing.
If you want all the angles around the unit circle again you count by 30.
So 0, 30, 60, 90, 120, 150, 180, somewhere around here 240... excuse me 210, 240, 270,
300, and somewhere around here 330 degrees.
Then you get back to 360.
These are all again degree marks.
Now that is only 12 angles.
You need to memorize 16 angles.
So the other angles are going to from my counting by 45 degrees.
So again we are going to count by 30, then we are going to count by 45 degrees.
Those are the big ones to remember.
These right here of 30 and 45.
We are going to count by 45 degrees since we just got done counting by 30.
0, 45, 90, and 135, 180, 225, 270, 315, and back to 360.
So those are our 16 angles on the unit circle that we must have memorized.
Now let's talk about the radian measures which are also must be memorized.
If you want to get all your unit circle quizzes done quickly and accurately and get those
easy points.
And I will quiz my kids to death on the unit circle quiz.
Hopefully your teachers are as well.
This is hugely important to memorize to make your work later a lot more efficient.
Ok.
We already know that 180 degrees is equal to pi radians.
Now I made this circle by counting by 30 and 45 degrees.
So let's start with 30 degrees.
How many 30 degrees are in 180?
1, 2, 3, 4, 5, 6.
So if this is pi radians and I divid by 6 to get the thirty, then 30 degrees in terms
of radians is pi/6.
So if we counted by 30's to get around the unit circle and make 12 of the 16 angles,
we are going to count by pi/6 to find them in radians.
Now I am also going to reduce these as we go.
So 1/6 and another 1/6, for another 30 degrees because again pi/6 is 30 degrees...
you really want to get that memorized, so 1/6, 2/6.
2/6 reduces to 1/3.
1/6, 2/6, 3/6 and that reduces to pi/2.
1/6, 2/6, 3/6, 4/6 reduces to 2pi/3 radians. 1/6, 2/6, 3/6, 4/6, 5pi/6.
We are just counting by 1 over 6.
6/6, 7/6.
Counting every 30 degrees.
7/6, 8pi/6... 8/6 is 4/3.
If you lose count go back to a denominator of 6 because that is the 30 degree denominator.
7/6, 8/6, 9/6, 10/6.
Again I am reducing this as I go if you are worried about why these denominators are not
6.
Then 7pi over... uh, 11pi/6.
Now, that is 12 of the radian measures you must have eventually memorized for the unit
circle.
To find the remaining ones we counted by 45 degrees.
So how many 45's are in 180?
Well, let's see.
1, 2, 3, 4.
180 divided by 4 is 45 degrees, so pi divided by 4 is our radian measure.
Now we are going to finish off getting our 16 angle measures by adding by pi/4.
So 1/4, 2/4, 3pi/4, 4/4, 5pi/4, 6pi/4 and we already have it... it is reduced, and finally
7pi/4.
These are the 16 degree measures and 16 radian measures that are around the unit circle.
This will eventually be filled up with the concepts of sine and cosine.
How much can I talk about this before I bore you to death and my time runs out.
Ok, a few minutes:)
If you notice these angles 30, and 60, and 45 and you think back in Geometry.
Those two speciality right triangles that you had to learn.
The 45-45 right triangle and the 60-30 triangle.
Right?!
Those triangles are going to help us set up these trig functions of sine and cosine.
Those two special triangles that you learned in Geometry, the 60-30 and the 45-45... those
ratios in the sides of those two speciality triangles are what is going to give us the
sine and cosine values, eventually in a later lecture, that we will need to memorize around
this unit circle.
If you are one of my students, you can build these equations...or these angles up counting
them out.
When we learn sine and cosine, you can use your special 60-30 and 45-45 triangles that
you learned in Geometry to build up the and sine and cosine values, however if you are
going to keep doing well in my class... if you are going to make your work more efficient
through the later stages of Trig and as you move into Calculus you must...you must...you
must remember or memorize all of the sine and cosine values around the unit circle.
All the sine and cosine values for those 16 angles once we get this entire concept of
the unit circle set up.
I am Mr. Tarrou.
That is your introduction to Standard Position Angles and Radian Measures.
BAM!!!
Go Do Your Homework:D