General Contents
Detailed Contents
Index
Polar to Rectangular 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.
We wish to convert the point
to rectangular coordinates.
General Contents
Detailed Contents
Index
First, let’s plot this point.
What fraction of a circle is
?
Since
, we need
of the full circle.
Plot the point on paper. Then check your work by clicking “Next”.
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Show the projections onto the
x
- and
y
-axes on your plot. Then check your work by clicking “Next”.
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How can we obtain the
x
-coordinate of this point?
We can use trig on the triangle.
How long is the hypotenuse?
4 units.
Does the projection on the
x
-axis involve the opposite or adjacent side?
The adjacent side.
Which trig function should we use?
The cosine.
Combine these ideas to express the
x
-coordinate.
For this point,
Express the
y
-coordinate.
State the coordinates of this point in (
x, y
) form.
The point with polar coordinates
is the same as the point with rectangular coordinates
.
Can we express this with just decimals?
Yes.
Do that.
The polar coordinates are (4, 0.52).
The rectangular coordinates are (3.46, 2.00).
Will this work for points in the other three quadrants?
Yes.
Even for the algebraic signs of the rectangular coordinates?
Yes.
How can that be?
In those quadrants the polar angle is greater than
,
and the trig functions have the correct algebraic signs.
Let’s try this for the point with polar coordinates (3.5, 2.7). In order to plot this, we need to estimate what fraction of a circle is represented by 2.7 radians. What is that fraction?
Since a full circle consists of 2
p
= 6.28 radians, we see that our 2.7 radians is more than
one third of a circle, but less than half.
Plot this point on paper. Then check your work by clicking “Next”.
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What algebraic sign do you expect for the
x
-coordinate?
Negative.
And for the
y
-coordinate?
Positive.
Do we still express the coordinates with the same trig functions?
Yes.
Set this up.
Evaluate these and express the rectangular coordinates in (
x, y
) form.
(-3.16, 1.49)
Do these results agree with the diagram?
Yes.
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to rectangular
coordinates.