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:)