# 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.

## 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 + √ / 4 = 0

to rectangular form.

Solution to Problem 3:

Rewrite the given equation as follows:
t + √ / 4 = 0
t = - √ / 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 and Links to Polar Coordinates and Trigonometry

Polar Coordinates.