To convert the polar equation r = 4 sin(θ) into rectangular coordinates, we will use the relationships between polar and rectangular coordinates:
- x = r cos(θ)
- y = r sin(θ)
- r = √(x² + y²)
Starting with the polar equation:
r = 4 sin(θ)
we can express sin(θ) in terms of rectangular coordinates:
- sin(θ) = y/r
Substituting this into the polar equation:
r = 4(y/r)
Next, we can multiply both sides by r to eliminate the fraction:
r² = 4y
Now, replace r² with x² + y²:
x² + y² = 4y
To rewrite this equation into a more recognizable form, we can rearrange it:
x² + y² - 4y = 0
Now, complete the square for the y terms:
- y² – 4y can be expressed as (y – 2)² – 4.
Therefore, substituting back into the equation gives:
x² + (y - 2)² - 4 = 0
Adding 4 to both sides results in:
x² + (y - 2)² = 4
This is now the equation of a circle with a center at (0, 2) and a radius of 2.
In conclusion, the transformation of the polar equation r = 4 sin(θ) into rectangular coordinates yields the equation of a circle in the form:
x² + (y - 2)² = 4