We are asked to convert
the Cartesian coordinates four comma six
to polar coordinates in the form r comma theta,
where r is greater than zero
and theta is between zero and two pi radians.
Remember, polar coordinates are not unique.
Let's begin by applying the point four comma six
on the coordinate plane.
From the origin we move right four units
and up six units.
R is the distance from the pole to the point,
and therefore the length of this segment here is r.
And then the angle theta is the angle of rotation
from the positive x-axis to the point,
which would be this angle here.
Now to help us determine r and theta,
let's sketch the reference triangle
in the first quadrant where r
is the hypotenuse of the right triangle.
So this would be one leg and this would be the other leg.
Now because x is four, let's label this leg four.
And because y is six, let's label this leg six.
Notice now we can see, we can find r,
by using the Pythagorean theorem,
which is why r is equal to the square root
of the quantity x squared plus y squared.
And since tangent theta is equal to the ratio
of the opposite side to the adjacent side,
we could say tangent theta is equal to y divided by x.
Solving for theta, we have theta
equals arctangent of y divided by x.
But we need to keep in mind though
that the output, or range, of arctangent
is the open interval from negative pi over two
to positive pi over two,
which means if the angle is not in the first quadrant,
the arctangent function value
is not going to give us the correct angle,
but it will give us the reference angle
that will help us determine the angle we want
in the desired 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 quantity four squared plus six squared,
which is equal to the square root of 16 plus 36.
16 plus 36 is equal to 52.
We have r is equal to the square root of 52.
This will simplify if 52
contains any perfect square factors.
To see if it does, let's look
at the prime factorization of 52.
52 is equal to two times 26.
26 is equal to two times 13.
Two times two, or four, is a perfect square.
And therefore, the square root of 52 does simplify.
52 is equal to four times 13,
and therefore we can write the square root of 52
as the square root of four
times the square root of 13.
Simplifying we have r is equal to two square root of 13.
And let's determine the angle theta,
where theta is equal to arctangent of y divided by x.
So in our case we have theta is equal to arctangent
of six divided by four,
which is equal to 3/2,
giving us theta is equal to arctangent of 3/2.
And now we go to the calculator.
We first want to make sure we are in radian mode.
So we press mode,
and notice radian is highlighted
so we are in radian mode.
We go back to the home screen.
Press second tangent for arctangent or inverse tangent,
three divided by two, close parenthesis and enter.
We're told to round to four decimal places.
Because we have a nine in the fifth decimal place,
we round up to 0.9828.
This angle is the angle we are looking for
because it is in the first quadrant
and in the interval from zero to two pi radians.
So the polar coordinates
for the Cartesian coordinates four comma six,
using the given conditions,
are two square root 13 comma 0.9828.
I hope you found this helpful.