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Hi everyone! Welcome back to integralcalc.com. Today we’re going to be doing another spherical
coordinates problem today. And the coordinates that we’re given are two, pie over three,
three pie over two. And, we’re going to be taking this spherical coordinates and converting
them into rectangular or Cartesian coordinates. So, couple of things to remember here, the
spherical coordinates come in the form rho, phi or phi, theta and we’re going to be
using these three formulas here to find x, y, and z of our Cartesian or rectangular coordinates.
So really as long as we can remember that our spherical coordinates are in this order
here rho, phi, and theta and if we can remember these three formulas and we’ve… I guess,
know how to use the unit circle, we’re… we should be home free with this kind of a
problem. So, the first thing that we’ll do is plug
in two for rho, pie over three for phi, and three pie over two for theta everywhere where
those variables occur in these formulas and what we’ll get are the following three formulas
for x, y, and z. Notice here that we plugged in two for rho, pie over three for phi, and
three pie over two for theta here which gives us our x equation, our y equation, and our
z equation and if we just simplify the three of these we’ll get our Cartesian or rectangular
coordinate. So, in order to evaluate these three equations or simplify them, we’ll
need to use our unit circle. Notice that we only have a couple different angles here,
we’re going to be looking for sine and cosine at the angle pie over three that occurs in
all three of our equations for x, y and z and we’re also going to be looking at three
pie over two. So, let’s jump over here to our unit circle and… and look for those
two angles. Remember, the first one we’re looking at is pie over three so we find pie
over three along our unit circle and at this angle we are looking for both cosine and sine.
Remember that when you’re using the unit circle, if you’re asked to find cosine of
an angle, you’ll be looking at the x coordinate there, if you’re asked to find sine of an
angle you’ll be looking at the y coordinate there. So, cosine of pie over three is one
half and sine of pie over three is the square root of three over two. So, we’ll keep those
two values in mind and then we’re also going to be looking at the angle three pie over
two. So if we continue along our unit circle here and we find three pie over two, here’s
the angle, we’re going to need cosine and sine of that angle, so again cosine of this
angle will be zero, the x coordinate, and sine of this angle will be negative one, the
y coordinate. So, given those four values, let’s head
back over here to our equations. If we bring in the values that we found in our unit circle,
sine of pie over three or the y coordinate where the angle is equal to pie over three
is the square root of three over two, cosine of three pie over two or the x coordinate
where the angle is equal to three pie over two is zero. So, when we multiply two by the
square root of three over two by zero of course we get zero. For y, we’re looking at the
y coordinate where the angle is equal to pie over three which is the square root of three
over two, we’re looking at the y coordinate where the angle is equal to three pie over
two and that is negative one, both of these values getting pulled from my unit circle.
So, when we multiply those together, the two’s will cancel because this is in the numerator
and this is the denominator and we’re left with simply negative square root of three.
Looking at our z equation here, we will substitute one half because that is where… that’s
the x coordinate where the angle is equal to pie over three, one half. So then, we multiply
one half by two and we get one. So, that’s all we have to do to simplify
our equations and to find our final answer. We combine all three of these x, y, and z
and we get zero, negative square root of three, one as our rectangular or Cartesian coordinate
and this is exactly equal to the spherical coordinate two, pie over three, three pie
over two. So I hope that helped you guys and I’ll
see you in the next video. Bye!