We're asked to convert
the Cartesian coordinates six comma negative two
to polar coordinates in the form r comma theta,
where r is greater than zero,
and theta is in the interval from zero to two pi radians,
closed on zero, open on two pi.
Let's begin by plotting the point on the coordinate plane.
Starting at the origin, we move right six units,
and then down two units.
This is the location of six comma negative two.
In polar coordinates, r is the distance
from the pole to the point,
and therefore, r is this distance here.
Next, because we want theta to be in the interval
from zero to two pi radians,
theta starts along the positive x-axis,
and then rotates counterclockwise to the point,
and therefore, this is the angle theta
we are looking for in polar coordinates.
For the next step, let's sketch the reference triangle
in the fourth quadrant where r is the hypotenuse,
and therefore, this is one leg and this is the other.
Because x is equal to six, we label this leg six.
Because y is negative two, we label this leg negative two.
Using the reference triangle and the Pythagorean theorem,
r is equal to the square root
of the quantity x squared plus y squared.
Remember, r can be positive or negative,
but we are told to use a positive value for r.
Then using the reference triangle, notice tangent theta
is equal to the ratio of the opposite side
to the adjacent side, and therefore,
tangent theta is equal to y divided by x.
Solving for theta, we have theta
equals r tangent over y divided by x.
But we do need to remember the output or range
of our tangent is the open interval
from negative pi over two to positive pi over two.
So, the arctangent function value is not going to give us
the angle we want in the interval from zero to two pi,
but it will give us the reference angle that we can use
to determine the given angle in the given interval.
Let's begin by determining r,
where again, r is equal to the square root
of the quantity x squared plus y squared,
which in our case, is the square root
of the quantity six squared plus the square of negative two,
which is equal to the square root
of the quantity 36 plus four.
R is equal to the square root of 40,
which will simplify because 40 does contain
the perfect square factor of four.
40's equal to four times 10,
and therefore, the square root of 40
is equal to the square root of four
times the square root of 10.
The square root of four is equal to two.
R simplifies to two square root 10.
We know this is completely simplified
because 10 does not contain any perfect square factors.
Now let's work on determining the angle theta,
where theta is equal to arctangent of y divided by x,
which in our case, is negative two divided by six.
Simplifying, we have arctangent of negative 1/3,
and now we go to the calculator.
It's important to make sure the calculator
is in radian mode.
Press mode.
Radian is highlighted.
Go back to the home screen.
Press 2nd + Tangent.
Negative one divided by three.
Close parenthesis, and enter.
Running to four decimal places
because we have a five in the fifth thousandth place.
The angle rounds to negative 0.3218.
But this is not the angle we're looking for
because we're looking for the angle theta
in the interval from zero to two pi radians.
So, the calculator just gave us the negative angle
in the fourth quadrant.
The calculator gave us this angle here.
Notice how this angle is co-terminal
with the angle we are looking for,
and therefore, to find the positive angle
in the interval from zero to two pi
that has the same tangent function value,
we need to add two pi radians to negative 0.3218 radians.
So, if we add two pi radians, we are then rotating
counterclockwise one full rotation from this terminal side,
which would give us the positive angle we are looking for.
And because we already had this value in the calculator,
we can simply press plus two pi.
Enter.
Running to four decimal places,
we have approximately 5.9614 radians.
Which means the polar coordinates
that satisfy the given conditions
for the Cartesian coordinates six comma negative two,
are two square root 10 comma 5.9614 radians.
I hope you found this helpful.