1.

Convert the equation r = 2 sin θ to cartesian coordinate​

Answer»

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.


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