We're given point in cartesian coordinates
and asked to find possible cylindrical coordinates.
So the given point has coordinates four comma negative three
comma negative five.
So we know x equals four, y equals negative three,
and z equals negative five.
To find possible cylindrical coordinates
we define r comma theta comma z.
But we are asked to use positive values for r and theta
and also to round to two decimal places if needed.
So to find r, theta, and z
we'll use our formulas here below.
But for a quick review r comma theta
is the polar representation of a point in the xy plane
where r is the directed distance from the origin
in the xy plane.
This distance here.
Theta is the angle counter clockwise from the polar axis
or positive x axis in the xy plane pictured here.
And finally z is the directed distance up or down
from r comma theta to the point,
which would be this directed distance here.
So going back to our example,
so we can better visualize this,
let's plot the point four comma negative three
in the xy plane.
So this would be the x axis, this would be the y axis.
And since the x coordinate is positive four
we'd go right four units,
and because the y coordinate is negative three
we go down three units.
So this would be the point four comma negative three
in the xy plane.
And therefore this distance here...
Would be r.
And then angle theta would be any angle that terminates
along this side.
But since we're looking for a positive value for theta
the least positive value for angle theta
would be this angle here.
Notice how if we wanted to we could also sketch out
a reference triangle.
Which would look like this.
Where this would be labelled four
and this would be labelled negative three.
So notice how to find r
we'll just be using the Pythagorean theorem
given here as r squared equals x squared plus y squared.
Let's begin by determining r.
We know r squared is equaled to x squared or four squared
plus y squared which is negative three squared.
So we have r squared is equal to 16 plus nine.
So r squared is equal to 25.
And then we'll take the square root
of both sides of the equation.
Notice here we'll have two solutions for r;
r equals plus or minus five.
We are asked to use a positive value for r though
so we'll use r equals five.
Remember if we use a negative value for r
then we plot the point in the xy plane
in the opposite direction of our angle theta.
Next to find theta we'll use the equation
tangent theta equals y divided by x.
So we have tangent theta is equal to y divided by x
would be negative three divided by four or negative 3/4.
And now to solve for angle theta
we would take the inverse tangent or arctangent
of both sides of the equation.
So theta is equal to inverse tangent of negative 3/4.
Here we might recognize that the output
of the inverse tangent function is on the open interval
from negative pi over two to positive pi over two.
So the calculator is not going to give us angle theta
that we're looking for.
But we can still use the calculator to find this angle.
Let's first verify that we are in radian mode.
So we'll press the mode key.
Radian is highlighted so go back to the home screen.
Press second tangent negative three divided by four
close parenthesis and enter.
So notice how the calculator is giving us an angle
that terminates here but it's a negative angle.
This would be approximately negative 0.64 radians.
Let's go ahead and sketch this.
The calculator gave us this angle here.
Our angle is coterminal to this
and therefore we can take this angle here
and add two pi radians to find
the least positive angle for angle theta.
Since we already have this value in the calculator
we can press plus two pi.
Enter.
So the angle theta pictured here in red
would be approximately 5.64 radians.
So again, the calculator gave us theta was approximately
negative 0.64 radians,
but to find the least positive value for angle theta
we added two pi radians to find our angle.
Which is approximately 5.64 radians.
So that's our second coordinate.
The third coordinate is the z coordinate;
which is the same as the z coordinate
of the cartesian point.
So z is equal to negative five.
So here is one possible way to express the given point
using cylindrical coordinates.
I hope you found this helpful.