Convert the equation r = 2 sin θ to cartesian coordinate
Answers
Step-by-step explanation:
When dealing with transformations between polar and Cartesian coordinates, always remember these formulas:
x
=
r
cos
θ
y
=
r
sin
θ
r
2
=
x
2
+
y
2
From
y
=
r
sin
θ
, we can see that dividing both sides by
r
gives us
y
r
=
sin
θ
. We can therefore replace
sin
θ
in
r
=
2
sin
θ
with
y
r
:
r
=
2
sin
θ
→
r
=
2
(
y
r
)
→
r
2
=
2
y
We can also replace
r
2
with
x
2
+
y
2
, because
r
2
=
x
2
+
y
2
:
r
2
=
2
y
→
x
2
+
y
2
=
2
y
We could leave it at that, but if you're interested...
Further Simplification
If we subtract
2
y
from both sides we end up with this:
x
2
+
y
2
−
2
y
=
0
Note that we can complete the square on
y
2
−
2
y
:
x
2
+
(
y
2
−
2
y
)
=
0
→
x
2
+
(
y
2
−
2
y
+
1
)
=
0
+
1
→
x
2
+
(
y
−
1
)
2
=
1
And how about that! We end up with the equation of a circle with center
(
h
,
k
)
→
(
0
,
1
)
and radius
1
. We know that polar equations of the form
y
=
a
sin
θ
form circles, and we just confirmed it using Cartesian coordinates.