We want to convert the point
two comma three pi divided by four
comma two pi divided by three,
which is given in spherical coordinates
to Cartesian coordinates or rectangular coordinates.
So we want to find x comma y comma z,
given that we know rho equals two,
theta equals three pi divided by four,
and phi equals two pi divided by three.
So we'll find the Cartesian coordinates
using our formulas provided here.
So x is equal to
rho times sine phi times cosine theta,
which would be two times
sine two pi divided by three
times cosine three pi divided by four.
And then y is equal to
rho times sine phi times sine theta,
which would be two times sine two pi divided by three
times sine three pi divided by four.
And finally z is equal to rho times cosine phi,
which is two times cosine two pi divided by three.
And now to find these trig function values,
we can use the unit circle or reference triangles.
Let's first use reference triangles,
and then we'll check the values on the unit circle.
So we'll sketch two pi divided by three
and three pi divided by four in standard position
to sketch the reference triangles.
So two pi divided by three is also equal to 120 degrees.
So the initial side would be here,
the terminal side would be approximately here,
where we'd have a reference angle
of pi divided by three radians, or 60 degrees.
So the reference triangle would look something like this,
where we have a 30 60 90 reference triangle.
So we can label the three sides one, two, square root three.
But because we're in the second quadrant
where x is negative, this would be negative one.
So we can use this reference triangle
to find the trig function values
for two pi divided by three radians.
And now we'll sketch three pi divided by four radians,
which is also equal to 135 degrees.
So the initial side would be here,
the terminal side would be in the second quadrant here,
where now the reference angle would be
pi divided by four radians, or 45 degrees.
So the reference triangle would look like this.
And now because we have a 45 45 90 reference triangle,
we can label the three sides one, one, square root two.
But again, because we're in the second quadrant,
this leg would be negative one.
So this would be the reference triangle
for three pi divided by four radians.
So now we can find our trig function values.
For x, we'd have two times
sine two pi divided by three would be equal
to the ratio of the opposite side to the hypotenuse,
which is square root three divided by two.
And then for cosine three pi divided by four,
that would be equal to the ratio
of the adjacent side to the hypotenuse,
which should be negative one divided by square root two.
But if we rationalize this, we should recognize
this would be negative square root two divided by two.
Let's go ahead and use
the cosine function value in this form.
So we'd have negative square root two divided by two.
Simplifying before we multiply,
two divided by two simplifies to one.
So x is equal to negative square root six divided by two.
And now for y, we have two times,
we already know sine two pi divided by three
is equal to square root three divided by two.
And now sine three pi divided by four is equal
to the ratio of the opposite side to the hypotenuse,
which would be positive one divided by square root two.
Well we know one divided by square root two
is also equal to square root two divided by two.
So let's go ahead and use square root two divided by two.
And notice here, again this simplifies.
So our product is positive square root six divided by two.
And then finally for z, we have two
times cosine two pi divided by three,
which would be two times,
cosine two pi divided by three is equal
to the ratio of the adjacent side to the hypotenuse,
which would be negative one divided by two, or negative 1/2.
So notice here, the z-coordinate is just negative one.
So the Cartesian coordinates are
negative square root six divided by two comma
positive square root six divided by two comma negative one.
And before we go, we did say we were going to verify
our trig function values on the unit circle.
Let's go ahead and do that.
Here's the angle two pi divided by three,
and here's the angle three pi divided by four.
So the x-coordinates give us the cosine function values,
and the y-coordinates give us the sine function values,
which does verify we found
our trig function values correctly.
I hope you found this helpful.