BAM!!! Mr. Tarrou. Ok, we did a video of helping you understand the polar coordinate system,
or attempting to. Hopefully that helped, if not, let me know what you still need me to
go over. Right now we are going to just a quick video for converting between rectangular
to polar and vise versa, from polar to rectangular. Going from polar to rectangular is a lot less
work and does not require nearly as much thought if you are just your calculator to get these
answers. However, from rectangular to polar, and I am going to kind of repeat the example
from our last video, it takes some thought. If you want a polar coordinate, you need an
r and a theta. So, r is equal to the square root, now my point down here is going to be
(-12,-5) which is in quadrant three, so we have the square root of negative twelve squared
plus negative five squared. That comes out to be the square root of, let's put it down
here, the square root of 144 plus 25 which is the square root of 169 or 13 units. This
point is 13 units away from the origin. x, y and r of the hypotenuse. You should know
5, 12, 13 as one of your pythagorean triples. I am just picking numbers that I can do relatively
easy in my head. Now, as far as finding the amount of rotation we must do from the positive
side of the x axis, if you will because what else am I going to call it, how much do you
have to rotate before you get the pointer...that theta...to go towards that coordinate of in
right now rectangular form (-12,-5). Well, ok we have x, we have y, we have r, we don't
really need this formula that textbooks generally give for conversions. Tangent of theta is
equal to y over x. We have known that now for a couple of chapters. And, when you know
x, y, and r you can use any trig function you want sine, cosine, tangent, secant, cosecant
if for some reason you want to use those. But this point is in quadrant three. So the
calculator, the value that comes out of a calculator is not going to be really very
usable. Inverse sine only gives us answers between negative pi over two and positive
pi over two, or negative 90 and positive 90. Inverse cosine as an inverse function only
gives us answers between zero and pi. Inverse tangent, as our textbooks like to recommend
that you use, again only gives answers between negative 90 and positive 90. So the fact that
this is in quadrant three means that you are not going to get a usable angle measure out
of calculator. Well, it will be usable but it will not be the final answer. So, let's
go ahead and stick with tangent anyway. So the tangent of theta which is equal to y over
x, is going to be negative five over negative twelve. Those of course cancel out. Theta
is equal to the inverse tangent of 5/12. The inverse tangent of 5/12 is approximately 22.6
degrees. Ok, great except that is in quadrant one. We need an angle in quadrant three. Again,
like your unit circle the...I don't know...how about cosine of 45 degrees or pi over four
is the square root of two over two. The cosine of 7pi/4 is the square root of two over two.
The reason why those two have the same answer, the cosine of 45 and the cosine of 315, or
the cosine of pi/4 and the cosine of 7pi/4, is because they have the same reference angle.
So we need an angle in quadrant three that has the same reference angle as this 22.6.
I am just using degrees because it is easier to do in my head. So, the actual theta that
we need is going to be pi, excuse me not radians, 180 degrees plus 22.6 degrees or 212.6 degrees.
If you were in radians you would be adding to pi. Now please, don't with these questions
keep going oh yeah we just add by 180, or to 180, or we add to pi, or we always subtract,
or we always use 360 or 2pi, please draw a picture of where you are at. If you are in
quadrant two, that is going to be 180 minus your reference angle or pi minus your reference
angle. If you are in quadrant three, now you have gone past the 180 or pi so you are going
to add your reference angle. If you are in quadrant four, you are not going to reference
pi or 180 at all, you are going to be looking at 360 degrees or 2pi. So draw a picture and
let yourself see where that point is so you correctly find the angle measurement that
you actually need. So, my final answer, my final polar coordinate is going to be instead
of (-12,-5), in polar form it is going to be (13,202.6) Now we can add 360, subtract
360, we could add 180 and make the radius negative. There is all kinds of things that
we can do to make other polar coordinates that describe that point in space. If you
need help with that, watch my Understanding Polar Coordinate System video. So, (-12,-5)
at least one possible answer in polar form is (13,202.6). I am doing this example again
because I gave an answer...I gave an answer...I gave a point in quadrant three and that means
the numbers out of our calculator are going to be usable but not the final answer for
the angle measurement. So pay attention to that as you do these problems. Alright! Let's
do this answer in reverse. What if we want to go from polar form to rectangular form?
Let's just use these same points again. Why not? We know they work. We want to go from
(13,202.6) and we want the rectangular form of that same point. We know it is (-12,-5)
but it is an issue, the fact that this point is in quadrant three? Are the numbers out
of the calculator going to be usable. Well, we have x, we have y, we have theta at 202.6.
We can use our basic understanding of trigonometry to find the x and the y coordinate and verify
that it is still (-12,-5). We have those skills, but when you do these conversions and you
are looking at a new section of the textbook, the author cannot just say well go back to
chapter four and look that up. So a little short cut formula, x is equal to r times cosine
theta and y is equal to r times sine theta. Now if I take this equation and I divide both
sides by r, don't I get the cosine of theta is x over r? If I divide both sides by r,
don't I get the sine of theta is equal to y over r? Have we not known that now for a
couple of chapters? So, there are no new formulas for these conversion problems. Ok, so this
is r, this is theta. We want x and y. So, x is going got be 13 (r) times the cosine
of theta or 202.6. Make sure that your calculator is in degree mode, or in radians if that is
what your textbook is doing, but I am using degrees. That should come out to be, well
of course we know it is going to be roughly negative twelve. I have already rounded off
in my scratch work, but it is going to be roughly negative twelve. Now why do I say
roughly. Because, you know, there is round off error. Did you take your decimal out to
the tenth place? 202.6........., no of course we rounded off a little bit. It is going to
be a little bit off due to round off error, but x is going to be negative twelve. Y is
equal to r times sine theta. That means that the y coordinate value out of your calculator
is going to be approximately, depending how much you rounded off, negative five. So we
have gone from polar to rectangular coordinates. Why is that we don't have to think about these
values? Because the cosine graph is a function. The cosine of zero is what? The cosine of
zero is one. So the cosine graph does this and it oscillates along the x axis. That is
a function. If you oscillate like this along the x axis along the x axis you are going
to pass the vertical line test. It is a function. So no matter what you put in there, whether
it is 30 degrees, 150 degrees, 240 degrees, or 202.6 degrees, you are going to get that
one answer you need because the cosine graph is a function and so is sine graph, the sine
function, they are functions. So, what ever we get out of our calculator is going to be
of course correct. That is not going to be the case with your inverse functions. We have
to restrict those answers so that it actually acts as a function. So, again inverse sine
your restricted values are only between positive ninety and negative ninety, inverse cosine
only gives you answers between zero and pi or 180, inverse tangent only gives answers
between negative pi/2 or -90 and positive pi/2 or 90. So, that is your conversions.
I am Mr. Tarrou. BAM!!! Go Do Your Homework:)