r
sin
θ
=
−
3
Explanation:
Imagine we have a point  
P
 with Rectangular (also called Cartesian) coordinates  
(
x
,
y
)
 and Polar coordinates  
(
r
,
θ
)
.
The following diagram will help us visualise the situation better:
https://keisan.casio.com/exec/system/1223526375
https://keisan.casio.com/exec/system/1223526375
We can see that a right triangle is formed with sides  
x
,  
y
 and  
r
, as well as an angle  
θ
.
We have to find the relation between the Cartesian and Polar coordinates, respectively.
By Pythagora's theorem, we get the result
r
2
=
x
2
+
y
2
The only properties we can say about  
θ
 are its trigonometric functions:
sin
θ
=
y
/
r
⇒
y
=
r
sin
θ
cos
θ
=
x
/
r
⇒
x
=
r
cos
θ
So we have the following relations:
⎧
⎪
⎨
⎪
⎩
r
2
=
x
2
+
y
2
y
=
r
sin
θ
x
=
r
cos
θ
Now, we can see that saying  
y
=
−
3
 in the Rectangular system is equivalent to say
r
sin
θ
=
−
3
Answer link
Jim G.
May 19, 2018
r
=
−
3
sin
θ
Explanation:
to convert from  
cartesian to polar
∙
x
x
=
r
cos
θ
 and  
y
=
r
sin
θ
⇒
r
sin
θ
=
−
3
⇒
r
=
−
3
sin
θ