BAM!!! Mr. Tarrou We are starting one of my favorite topics which is trigonometry. The
study of triangles and their rotations around a circle. This is used in engineering, measurements
of real life items that are too big to do with a ruler and a straight edge and stuff
like that. It has to do a lot with mathematics of rotation and spheres. We live on a globe.
A lot of mechanical items have tons of spinning items in them, just think of an engine with
a spinning crank shaft. So trig is very important in later studies of mathematics and engineering.
Let's get started with a very basic idea and some definitions. We are going to introduce
standard position angles and try to in probably a two part video explain what radian measures
are...that is another way of measuring rotation besides degrees. A standard position angle
has a vertex on the origin and it's initial side on the positive side of the x axis. So
if I wanted to start drawing a standard position angle, I would start off by putting the vertex
on the origin and initial side on the positive x axis. Now the terminal side is the only
part that is allowed to move in a standard position angle. So if you wanted to draw a
45 degree angle...BAM!! This is the terminal side. Well that is fine but that is an incomplete
picture. This is because one of the other definitions I have to talk about is coterminal
angles. Those are angles that have a measurement, two angles that have measurements that are
separated by a full rotation of either 360 degrees or 2pi radians. You might not know
what that means yet (radians), but hopefully by the end of this video you will. Ok, so
we have an initial side, a terminal side, now I need to add an extra part to this diagram
to indicate what direction the rotation is and how much rotation there is. So I am going
to draw a little arc that is going to indicate counter clockwise rotation and i did say that
we were drawing a 45 degree angle. Well what if I wanted to draw a negative 45 degree angle?
What would that negative rotation look like? So I am going to draw another angle on this
exact same picture. Well all angles in standard position have their vertex on the origin,
they all share the same initial side, so there that is the first part of my new angle I am
drawing. But, now I want to rotate negative 45 degrees. Well positive rotation is counter
clockwise from the positive side of the x axis, the initial side, and so I think you
might guess what negative 45 degrees is...that is clockwise rotation. Again from the positive
side of the x axis, so something like that. So there we have an initial and terminal side.
We have another initial and terminal side of the negative 45 degree angle. Ok, so there
you go! All drawings of standard position angles have a vertex on the origin, positive
side of the x axis, rotate counter clockwise for positive rotation and clockwise for negative
rotation. Now later on in this textbook and in your class you are probably studying bearing
questions. You know how to read a compass? Compass rotations, bearings, are going to
based off of due North so that might be a little bit confusing for some of you when
we get to those questions about bearing. But, a standard position angle, all of those rotations
do start off the positive side of the x axis. These are not like navigation angles. Now
you see I have got I, II, III, and IV up here. This is just the coordinate plane. This is
quadrant 1, this is quadrant ii, quadrant iii, and quadrant IV. So our 45 degrees is
in quadrant I and negative 45 degrees is in quadrant IV. Now let's talk about this idea
of coterminal angles having angles separated by full rotations of 360 degrees. I want to
draw an angle that is coterminal to 45 degrees. What is that going to look like? It is going
to be in standard position so it is going to share the same initial side that ALL standard
position angles share the same initial side. ALL of them have their vertex on the origin.
So you know this does not say co-initial angle, it says coterminal angle. Co-habitate, coexist,
cooperate...work together or share...they are sharing the same terminal side. That is
the big deal, that is the side that moves. They ALL share the initial side in standard
position. So, an angle that is coterminal to 45 degrees is going to look like a 45 degree
angle but that is not a complete picture. Is that a 45 degree angle or is that an angle
that is coterminal to a 45 degree angle? We don't know that without these little arcs
to indicate direction of rotation and how much. I need to rotate and additional full
rotation of 360 and I am going to show that by doing this. Start at the positive side
of the x axis, rotate positively because I want to do...I just decided to do a positive
rotation, and I am going to rotate all the way around until I get back to that same terminal
side. Now how many degrees have I graphed here? I have graphed a full 360 degrees plus
an additional 45 degrees to make an angle that is 360 plus 45 is 405 degrees. Now we
could also rotate in the opposite direction as well. We could instead of adding 360 like
I have done, we could subtract 360 and that will show an angle of rotation instead of
going counter clockwise, we would go clockwise. But if I do any more arcs on this you will
not be able to read anything. We could do 45 minus 360 and get a negative, what would
that be negative 315 degrees. You can add or subtract a full rotation and get a coterminal
angle. I have one more definition up here for you. It is a quadrantal angle. I have
drawn an angle in quadrant 1 and I have drawn an angle in quadrant 4 with the negative rotation.
If I draw an angle such as let's say 90 degrees, if I rotate exactly 90 degrees I am landing
the terminal side on an axis line. I am not actually in quadrant 2 and I am not in quadrant
1, I am exactly in between, this angle is not in a quadrant it is a quadrantal angle.
Any angle with a measurement of 90 or an integer multiple of 90 is going to be considered a
quadrantal angle. It's terminal side is not going to be in a quadrant at all. Ok, let's
erase all of this and get on to the next topic which is how to measure rotation with radians
and not degrees. We care about another way measuring rotation because in certain areas
of mathematics, where we are doing mathematics where things are spinning a lot like engineering,
if we are doing navigation around a globe a round sphere, some mathematics can be done
easier with radian measures instead of degrees. Now let's talk about a full rotation of 360
degrees. That is not too bad. We have a radius. So we have a circle, I have rotated 360 degrees.
Thus I have got a bottom heavy circle here. So let's talk about a measure of rotation
that we are comfortable with which is degrees and see if we can't convert it into the idea
of radians. For radians I want to talk about this idea of pi. Every time you see the letter
pi which stands for approximately 3.14, that does not mean that automatically you have
a radian measure. But pi of 3.14 does have a lot to do with radians. Do you know where
3.14 comes from? A lot of my students do not. Well, if you take a circle such as we have
here only maybe yours is actually round:) I am going to have this length of r and I
am going to try my best to estimate this length and wrap it around my not so perfect circle.
So maybe somewhere around here and that is one length of the radius. And so we have one
length of that radius. Let's estimate that the best we can again and say it is probably
somewhere around there and lay down another length of the radius of this circle. Now we
are at 2 radiuses. And then finally let's do another length of the radius which is probably
about there. Now we are at 3 radiuses. Where are we on the circle? We are almost to the
other side of the circle. So in a measure of rotation that we are comfortable with which
is degrees, I have just about gotten to the side of the circle...I have just about rotated
to 180 degrees. Now I am counting off the radiuses and I am almost at that 180 degrees.
What do you think is left in that little bit of a gap? What value do you think is in that
little piece right there? That little piece is approximately equal to .14 radiuses. So
3.14 is the number of radiuses, or radii, to get half way around a circle. Now you can
also say that pi 3.14 is the ratio of the entire circumference with the diameter. 3.14
diameters gets you all the way around the circle, but all of our formulas are going
to be based off the radius and not the diameter. So, I am going to say, and it is true, that
3.14 is the number of radii to get you half way around the circle. Half way around the
circle is 180 degrees, so 180 degrees is equal to pi radians. Now where do think they got
this idea, this name of radians from? Well because it is measured and based of the radiuses.
So radian is based off of the measure of the radius of the circle, and so radians is found
by doing s over r. S stands for arc length and dividing that by the length of the radius.
So regardless of how big the circle is, there is only one shape that is a perfect circle,
so regardless of the size of that circle 3.14 radiuses is what it takes to get you half
way around the circle. Pi radians equals 180 degrees. Nice:D Ok, you also might note that
1 radian gets you somewhere near the end of quadrant 1. 2 radians gets you into quadrant
2. 3 radians is still going to be in quadrant 2, almost getting you to 180 degrees. I am
getting close to running out of time so in the next lesson we are going to look at converting
from degrees to radians and building up all the angles around what you will know eventually...not
in this video...but what you will know eventually as the unit circle. That is hugely important
for you to memorize in the rest of your studies of Trig. I am Mr. Tarrou. I will be right
back:)