We're asked to convert the Cartesian coordinates
negative four comma negative four 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 first plot the point on the coordinate plane.
From the origin, we move left four units and then
down four units.
This is the point we want to write using polar coordinates
using the given conditions.
R is the distance from the pole to the point.
And therefore the length of this segment here is r.
And then theta is the angle starting along the positive
x axis, rotating to the point,
which would be this angle here, the angle from zero
to two pi radians.
To help us determine r and theta, let's now sketch
the reference triangle in the third quadrant
where r would be the hypotenuse of the reference triangle.
So this would be one leg and this would be the other.
Because x is negative four, we label this leg negative four.
Because y is also negative four, we label
this leg negative four.
Using the right triangle in the Pythagorean theorem,
r is equal to the square root of the quantity x squared
plus y squared.
And then using the reference triangle, tangent theta
is equal to the ratio of the opposite side
to the adjacent side or tangent theta is equal
to y divided by x.
Solving for theta, we have theta equals arc tangent
over y divided by x.
We need to be careful when determining the angle
theta though because the output of range of our tangent
is the open interval from negative pi over two
to a positive pi over two.
So the arc tangent function and value we get
from the calculator is not going to give us the angle
we want in the third quadrant but it will give us
the reference angle that we can use to determine
the angle we want in the third quadrant.
Let's begin by determining r where again r is equal
to the square root of the quantity x squared
plus y squared or in our case r is equal to the square root
of the square of negative four plus the square
of negative four.
Simplifying, we have the square root of 16 plus 16.
16 plus 16 is equal to 32.
R is equal to the square root of 32, which we'll simplify
because 32 does contain the perfect square factor of 16.
32 is equal to 16 times two.
And therefore the square root of 32 is equal
to the square root of 16 times the square root of two.
The square root of 16 is equal to four.
R simplifies to four, square root of two.
And now let's determine the angle theta.
Theta is equal to arc tangent of y divided by x,
which in our case is negative four divided by negative four.
Simplifying, we have arc tangent of one.
And we should be able to recognize an angle
that has a tangent function value of one.
This should remind us of a 45, 45, 90 reference triangle
where a tangent 45 degrees or a pi over four radians
is equal to one.
So if we recognize that, we know the calculator's going
to give us pi over four radians.
Let's assume we don't recognize this
and also show the calculator.
But the calculator is only going to give us
a decimal approximation.
But right away you might recognize that pi over four radians
is in the first quadrant and therefore this is not
the angle we are looking for in the third quadrant.
But let's first find theta on the TI-84.
We need to verify we are in radian mode.
Press mode, radian is highlighted.
Go back to the home screen.
Press second, tangent, one, close parentheses, and enter.
Because we have a nine in the fifth decimal place,
we round up to 0.7854 radians.
So the angle given by the calculator is this angle here
in the first quadrant in the opposite direction.
But remember, tangent is positive in the first
and third quadrants and therefore these two angles
have the same reference angle.
So the reference angle for the angle theta
we are looking for is pi over four radians
or approximately 0.7854 radians, which means to find
the angle theta in the given interval, we need to add
pi radians to pi over four radians or to the approximation.
So the exact angle theta would be pi over four
plus pi or 5/4 pi radians or the decimal approximation.
Going back to the calculator and adding pi
would be approximately 3.9270 radians.
So the exact polar coordinates that satisfy
the given conditions for the Cartesian coordinates
negative four comma negative four would be four
square root of two comma 5/4 pi radians.
But our homework system will also accept four
square root of two comma 3.9270 radians.
I hope you found this helpful.