# Converting Equations from Polar to Rectangular

Math Analysis - Polar Equation to Rectangular Equation

Tip: This isn't the place to ask a question because the teacher can't reply.

1 of 2 videos by AJ Speller

## Key Questions

As discussed below.

#### Explanation:

Polar coordinates are a complementary system to Cartesian coordinates, which are located by moving across an x-axis and up and down the y-axis in a rectangular fashion. While Cartesian coordinates are written as (x,y), polar coordinates are written as (r,Î¸).

In polar coordinates, a point in the plane is determined by its distance r from the origin and the angle theta (in radians) between the line from the origin to the point and the x-axis

The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction

Why is it called Polar form?
In polar coordinates the origin is often called the pole. Because we aren't actually moving away from the origin/pole we know that . However, we can still rotate around the system by any angle we want and so the coordinates of the origin/pole are .

Who invented polar coordinates?
The polar coordinate system is an adaptation of the two-dimensional coordinate system invented in 1637 by French mathematician RenÃ© Descartes (1596â€“1650). Several decades after Descartes published his two dimensional coordinate system, Sir Isaac Newton (1640â€“1727) developed ten different coordinate systems.

See explanation

#### Explanation:

Imagine an arm where one end is pinned at the centre of a circular style graph and at the other end is a pencil. Also, the length of this arm can change such that its length is governed by a formula involving the angle of rotation.

Suppose the length of the arm is $r$ and we had one equation that was $r = 5 \sin \theta$ and a second where $r = 5 \sin \left(3 \theta\right)$

Then we would have:

If you so chose you could graph it on standard graph paper. In which case it would look like:

• To convert an equation given in polar form (in the variables $r$ and $\theta$) into rectangular form (in $x$ and $y$) you use the transformation relationships between the two sets of coordinates:
$x = r \cdot \cos \left(\theta\right)$
$y = r \cdot \sin \left(\theta\right)$

You have to remember that your equation may need some algebraic/trigonometric manipulations before being transformed into rectangular form; for example, consider:

$r \left[- 2 \sin \left(\theta\right) + 3 \cos \left(\theta\right)\right] = 2$
$- 2 r \sin \left(\theta\right) + 3 r \cos \left(\theta\right) = 2$

Now you use the above transformations, and get:

$- 2 y + 3 x = 2$
Which is the equation of a straight line!

A more complicated situation can be the following example:
$\theta + \frac{\pi}{4} = 0$
You can write:
$\theta = - \frac{\pi}{4}$
Take the tangent of both sides and multiply and divide by $r$ the left side:
$\frac{r}{r} \cdot \tan \left(\theta\right) = \tan \left(- \frac{\pi}{4}\right)$
$\frac{r \sin \left(\theta\right)}{r \cos \left(\theta\right)} = - 1$
Transforming you get:
$\frac{y}{x} = - 1$
$y = - x$

• This key question hasn't been answered yet.

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