Convert the point negative two
comma negative one comma five to spherical coordinates.
Because the given point has Cartesian coordinates
or rectangular coordinates we know x equals negative two,
y equals negative one, and z equals five.
To find the spherical coordinates we need to find rho
comma theta comma phi using our formulas here below.
But for a quick review, rho is the distance
between the point and the origin.
This distance here.
Theta is the angle counter clockwise from the pole
or positive x axis in the xy plane.
This angle here.
This is the same angle we find in polar coordinates
in two dimenstions.
And phi is the angle between the positive z axis
and the point which is this angle here.
So going back to our example, let's first find rho
using our formula here.
So rho squared is equal to x squared plus y squared
plus z squared which would be negative two squared
plus negative one squared plus five squared.
So we have rho squared is equal to four plus one plus 25.
So rho squared is equal to 30.
So now to find rho we take the square root
of both sides of the equation.
Algebraically we'll have two solutions to this equation.
Where rho is equal to plus or minus the square root of 30.
But we always use a positive value
for rho so our first coordinate is square root 30.
Notice how from the directions,
we could find a decimal approximation for the square root
of 30 and then round to one decimal place,
but let's go ahead and give the exact value.
Now we'll find our angle theta.
We need to be careful, though,
to make sure we find angle theta in the correct quadrant.
To make sure we do this, let's plot the point negative two
comma negative one in the xy plane.
So we'd go left two units and down one unit
which means our angle theta must terminate
along this side here.
Now, of course, there's an infinite number of angles
that terminate along this side here, but we are asked
to find a positive value for angle theta.
Let's find the least positive value for angle theta
which would be this angle here.
And tangent theta is equal to y divided by x.
So tangent theta is equal to negative one divided
by negative two which equals positive one half.
So tangent theta is equal to positive one half
then theta would be equal to arc tangent or inverse tangent
of one half.
And now we'll go to the calculator.
Let's first make sure we are in radian mode.
So we'll press the mode key.
Notice radian is highlighted, so we'll go back
to the home screen.
And now we'll press second tangent for inverse tangent
or arc tangent one divided by two close parenthesis
and enter.
Now notice how the calculator is giving us an angle here
that terminates in the first quadrant not
in the third quadrant.
But we can use this angle to find the angle we're looking
for in the third quadrant.
So let's go ahead and sketch this angle first
which is approximately 0.4636 radians.
So the calculator gave us this angle here
which is approximately 0.4636 radians.
Well the angle that has the same tangent function value
as this angle here in the third quadrant
would have the same reference angle
so our angle has a reference angle of 0.4636 radians.
Which means our angle theta is going to be equal to
or approximately equal to the reference angle
of 0.4636 plus have a rotation or plus pi radians.
Let's also record this down here.
The calculator gave us approximately 0.4636
but our angle theta in the third quadrant
that has the same tangent function value
would be this angle plus pi radians.
So let's go ahead and go back to
the calculator one more time.
And since this angle was our last entry,
we can press plus pi.
So our angle theta in the third quadrant
is approximately 3.6052 radians.
And again following the directions from the question,
rounding to one decimal place.
We'd say theta is approximately 3.6 radians.
And now let's find phi or cosine phi is equal
to z which is positive five divided by the square root
of x squared plus y squared plus z squared
which is just rho which would be the square root of 30.
So if cosine phi is equal to five divided
by square root of 30.
Then phi is equal to inverse cosine
or arc cosine of five divided by square root of 30.
So now we'll go back to the calculator
and we'll enter second cosine five divided
by square root 30, right arrow, close parenthesis,
and enter.
So phi is approximately 0.4205 radians.
And for our homework we'll enter approximately 0.4 radians.
So here are spherical coordinates for the given point
in Cartesian coordinates.
I hope you found this helpful.