Equations in polar form are converted into rectangular form, using the relationship between polar and rectangular coordinates. Problems with detailed solutions are presented.
Problem 1: Convert the polar equation
R = 4 sin t
to rectangular form.
Solution to Problem 1:

We multiply both sides by R
R = 4 sin t
R^{ 2} = 4 R sin t

We now use the relationship between polar and rectangular coordinates: R^{ 2} = x^{ 2} + y^{ 2} and y = R sin t to rewrite the equation as follows:
x^{ 2} + y^{ 2} = 4 y
x^{ 2} + y^{ 2}  4 y = 0

It is the equation of a circle.
Problem 2: Convert the polar equation
R (2 sin t + 3 cos t) = 2
to rectangular form.
Solution to Problem 2:

Expand the left side of the given equation.
R(2 sin t + 3 cos t) = 2
2 R sin t + 3 R cos t = 2

Use y = R sin t and x = R cos t into the given equation to rewrite as follows:
2 y + 3 x = 2

The above is the equation of a line.
Problem 3: Convert the polar equation
t + Pi / 4 = 0
to rectangular form.
Solution to Problem 3:

Rewrite the given equation as follows:
t + Pi / 4 = 0
t =  Pi / 4

Take the tangent of both sides:
tan t = 1

Use y = R sin t and x = R cos t to write:
tan t = sin t / cos t = R sin t / R cos t = y / x

Hence:
y / x = 1
y =  x

The above is the equation of a line.
More references on polar coordinates and trigonometry topics.
Polar Coordinates.
