General Contents
Detailed Contents
Index
Rectangular to Polar Coordinates
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If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.
Convert the point given by (3, 2) in rectangular coordinates to polar coordinates.
General Contents
Detailed Contents
Index
First plot the point on paper and show its projections on the
x
- and
y
-axes.
Then check your graph by clicking “Next”.
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Show the polar coordinates
R
and
q
on the diagram.
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How can we find
R
.
We can use the Pythagorean Theorem on the
x
- and
y
-coordinates.
Set this up in terms of
R, x,
and
y
.
Apply this to the data.
How can we
find
q
?
We can use the right triangle and set up the trig relationship
for
q
.
Do it.
Substitute the data.
We get
Solve for
θ
.
θ
= 0.59 radians
Does
this value of
θ
agree
with the diagram?
Yes, it is just a little smaller than pi/4 = 45 deg, which would have a tangent value of 1.
Summarize this case by writing the polar coordinates for the rectangular point (3, 2).
The point with rectangular coordinates (3, 2) is also described in polar coordinates as (3.61, 0.59).
Will
this work
if the point is not in the first quadrant?
Yes.
Try it for the rectangular point (-5, 2).
First find the value of
R
.
Find tan
θ
.
Which quadrant can this be in?
Based only on this negative value,
θ
could be in either quadrant II or IV.
How do we decide?
Since the
x
-coordinate is negative and the
y
-coordinate is positive, the point must be in quadrant II.
Find
arctan
(
-0.4).
What quadrant is this in?
Quadrant IV.
How do we find the angle in quadrant II which also has -0.4 for its tangent?
We add
π
radians.
Do that.
We get
radians, or 158 degrees.
Plot the point and check these values.
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Do our values agree with the diagram?
Yes, R is a little more than 5, and theta is about 30 degrees less than 180 degrees.
Summarize this case by writing the polar coordinates of the point (-5, 2).
The point with rectangular coordinates (-5, 2) is also described in polar coordinates as (5.4, 2.76).
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
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