Convert Equation from Polar to Rectangular Form

Equations in polar form are converted into rectangular form, using the relationship between polar and rectangular coordinates. Problems with detailed solutions are presented.
In what follows the polar coordinates of a point are (R , t) where R is the radial coordinate and t is the angular coordinate.
The relationships between the rectangualr (x,y) and polar (R,t) coordinates of a points are given by
R 2 = x 2 + y 2       y = R sin t       x = R cos t

Problems on Converting Equation from Polar to Rectangular Form

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 + π / 4 = 0

to rectangular form.

Solution to Problem 3:

Rewrite the given equation as follows:
t + π / 4 = 0
t = - π / 4
Take the tangent of both sides:
tan t = tan (π / 4) = -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 and Links to Polar Coordinates and Trigonometry

Polar Coordinates.