Converting Polar Coordinates Example 5

Hi everyone. Welcome back to integralcalc.com. Today we're going to do another example of
how to convert polar coordinates to Cartesian coordinates. The polar coordinates that we're
given in this problem are 2, negative pi/4. So the formulas that we're going to use to
convert these polar coordinates to Cartesian coordinates are the following.
We are going to use these two formulas to take these polar coordinates and convert them
into Cartesian form or rectangular form. So, we are going to first recognize that our polar
coordinates are in the form r, theta. Normally with rectangular or Cartesian coordinates,
the coordinates are x, y. But when you're talking about polar coordinates,
the x-coordinate is r and the y-coordinate is theta.
r being the distance that you go out from the origin, theta being the angle from the
x-axis. So anyway, 2 is r and negative pi/4 is theta
which means we’re going to plug those into our two formulas here for r and theta. The
equation we're going to get here to find our x-coordinate, plugging in x for r and negative
pi/4 for theta gave us 2 cos of negative pi/4. Similarly, when we plug those same two numbers
in to our equation for y, we get y equals 2 sin of negative pi/4.
That is how we set up our two equations to find our rectangular coordinates. To evaluate
cos of negative pi/4 and sin of negative pi/4, we are going to need to use our Unit Circle.
You could do this two ways. You could either take the angle negaitve pi/4 and start at
the angle 0 on the unit circle and work backwards an angle of negative pi/4 or you could convert
this negative pi/4 to a postive angle. The way that you do that, remember that the angle
around the unit cricle, 360 degrees in other words is 2pi. So, if you say 2pi and then
you add negative pi/4, you'll end up with a positve angle.
So when you find the lowest common denominator, you're multiplying 2pi by 4/4 so you get 8pi/4
minus pi/4 and when you simplify that, you get 7pi/4.
What this tells you is that negative pi/4 is the same angle as 7pi/4.
It’s going to be the same angle on the unit circle and you can evaluate either way. So
let's go ahead and jump over to our Unit Circle here. You can see the angle 7pi/4. We converted
our negative pi/4 and we found 7pi/4 so you can evaluate here.
But if you didn't want to go through the process of converting your angle, you could start
from the angle 0 right here and move clockwise along the unit circle until you reach negative
pi/4. Normally, we move in a counter-clockwise direction but since, it's a negative, we move
the other way around. In this case you can see the denominator of this first angle here
is a 6 and then the denominator is a 4 and then a 3. Since we have negative pi/4 and
4 is our denominator, we're looking for the first angle with a denominator of 4 which
is this angle right here. That's going to land you right here at 7pi/4.
Notice that if you added 7pi/4 and pi/4, you would get 8pi/4 which would give you 2pi and
land you right back here. So, negative pi/4 is the same thing as 7pi/4. But either way,
we're looking for both cos of this angle and sin of this angle. Remember that when you're
looking at the unit circle, cos of an angle means that you're looking at the x-coordinate.
sin of an angle means you're looking at the y-coordinate.
Let's head back over here to our equations. You can see that we have cos of negative pi/4.
We already found that the cos of that angle is positive square root of 2 over 2.
With the y equation sin of negative pi/4 the y-coordinate at negative pi/4 or 7pi/4 was
negative square root of 2/2. So we plug those in for cos negative pi/4
and sin of negative pi/4. When we simplify this, we get the square root
of 2 in the numerator and 2 in the denominator and our x-coordiante is going to be square
root of 2. Same with y here, we get the 2 in the numerator
and in the denominator to cancel and we're left with negative square root of 2.
When we put these two together, these x and y-coordiantes, our final answer is square
root of 2, negative square root of 2, which is exactly equivalent in rectangular or Cartesian
coordinates to our polar coordiante of 2, negative pi/4.
So I hope that helped you guys and I'll see you in the next video. Bye.