Polar and Rectangular Conversions

You are hiking one day with friends. When you stop to examine your map, you mark your position on a polar plot with your campsite at the origin, like this

You decide to plot your position on a different map, which has a rectangular grid on it instead of a polar plot. Can you convert your coordinates from the polar representation to the rectangular one?

Converting Polar Coordinates to Rectangular Coordinates

Just as $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ are usually used to designate the rectangular coordinates of a point, $\begin{align*}r\end{align*}$ and $\begin{align*}\theta\end{align*}$ are usually used to designate the polar coordinates of the point. $\begin{align*}r\end{align*}$ is the distance of the point to the origin. $\begin{align*}\theta\end{align*}$ is the angle that the line from the origin to the point makes with the positive $\begin{align*}x-\end{align*}$axis.

The diagram below shows both polar and Cartesian coordinates applied to a point $\begin{align*}P\end{align*}$. By applying trigonometry, we can obtain equations that will show the relationship between polar coordinates $\begin{align*}(r, \theta)\end{align*}$ and the rectangular coordinates $\begin{align*}(x, y)\end{align*}$

The point $\begin{align*}P\end{align*}$ has the polar coordinates $\begin{align*}(r, \theta)\end{align*}$ and the rectangular coordinates $\begin{align*}(x, y)\end{align*}$.

Therefore

$$\begin{align*}x& = r \cos \theta && r^2 = x^2+y^2\\ y&= r \sin \theta && \tan \theta = \frac{y}{x}\end{align*}$$

These equations, also known as coordinate conversion equations, will enable you to convert from polar to rectangular form.

Converting Coordinates 

Given the following polar coordinates, find the corresponding rectangular coordinates of the points: $\begin{align*}W(4,-200^\circ),H \left (4, \frac{\pi}{3} \right )\end{align*}$

For $\begin{align*} W(4,-200^\circ), r = 4 \end{align*}$ and $\begin{align*}\theta = -200^\circ\end{align*}$

$$\begin{align*}x & = r \cos \theta && y = r \sin \theta\\ x &= 4 \cos (-200^\circ) && y = 4 \sin(-200^\circ)\\ x &= 4(-.9396) && y = 4(.3420)\\ x & \approx - 3.76 && y \approx 1.37\end{align*}$$

The rectangular coordinates of $\begin{align*}W\end{align*}$ are approximately $\begin{align*}(-3.76, 1.37)\end{align*}$.

For $\begin{align*}H \left ( 4,\frac{\pi}{3} \right ), r = 4\end{align*}$ and $\begin{align*}\theta = \frac{\pi}{3}\end{align*}$

$$\begin{align*}x &= r \cos \theta && y = r \sin \theta\\ x &= 4 \cos \frac{\pi}{3} && y = 4 \sin \frac{\pi}{3}\\ x &= 4 \left ( \frac{1}{2} \right ) && y = 4 \left ( \frac{\sqrt{3}}{2} \right )\\ x &= 2 && y = 2 \sqrt{3}\end{align*}$$

The rectangular coordinates of $\begin{align*}H\end{align*}$ are $\begin{align*}(2, 2 \sqrt{3})\end{align*}$ or approximately $\begin{align*}(2, 3.46)\end{align*}$.

Converting Equations 

1. In addition to writing polar coordinates in rectangular form, the coordinate conversion equations can also be used to write polar equations in rectangular form.

Write the polar equation $\begin{align*}r = 4 \cos \theta \end{align*}$ in rectangular form.

$$\begin{align*}r &= 4 \cos \theta\\ r^2 &= 4r \cos \theta && Multiply \ both \ sides \ by \ r.\\ x^2 + y^2 &= 4x && r^2 = x^2 + y^2 \ and \ x = r \cos \theta\end{align*}$$

The equation is now in rectangular form. The $\begin{align*}r^2\end{align*}$ and $\begin{align*}\theta\end{align*}$ have been replaced. However, the equation, as it appears, does not model any shape with which we are familiar. Therefore, we must continue with the conversion.

$$\begin{align*}x^2 - 4x + y^2 &= 0\\ x^2 - 4x + 4 + y^2 &= 4 && Complete \ the \ square \ for \ x^2 - 4x.\\ (x - 2)^2 + y^2 &= 4 && Factor \ x^2 - 4x + 4.\end{align*}$$

The rectangular form of the polar equation represents a circle with its centre at (2, 0) and a radius of 2 units.

This is the graph represented by the polar equation $\begin{align*}r = 4 \cos \theta\end{align*}$ for $\begin{align*}0 \le \theta \le 2 \pi\end{align*}$ or the rectangular form $\begin{align*}(x - 2)^2 + y^2 = 4.\end{align*}$

2. Write the polar equation $\begin{align*}r = 3 \csc \theta\end{align*}$ in rectangular form.

$$\begin{align*}r &= 3 \csc \theta\\ \frac{r}{\csc \theta} &= 3 && divide \ by \csc \theta\\ r \cdot \frac{1}{\csc \theta} &= 3\\ r \sin \theta &= 3 && \sin \theta = \frac{1}{\csc \theta}\\ y &= 3 && y = r \sin \theta\end{align*}$$

Examples

Example 1

Earlier, you were asked to convert your coordinates from polar representation to the rectangular one. 

You can see from the map that your position is represented in polar coordinates as $\begin{align*}(3,70^\circ)\end{align*}$. Therefore, the radius is equal to 3 and the angle is equal to $\begin{align*}70^\circ\end{align*}$. The rectangular coordinates of this point can be found as follows:

$$\begin{align*}x & = r \cos \theta && y = r \sin \theta\\ x &= 3 \cos (70^\circ) && y = 3 \sin(70^\circ)\\ x &= 3(.342) && y = 3(.94)\\ x & \approx 1.026 && y \approx 2.82\end{align*}$$

Example 2

Write the polar equation $\begin{align*}r = 6 \cos \theta\end{align*}$ in rectangular form.

$$\begin{align*}r & = 6 \cos \theta \\ r^2 & = 6r \cos \theta \\ x^2 + y^2 & = 6x \\ x^2 - 6x + y^2 & = 0 \\ x^2 - 6x + 9 + y^2 & = 9 \\ (x - 3)^2 + y^2 & = 9\end{align*}$$

Example 3

Write the polar equation $\begin{align*}r \sin \theta = -3\end{align*}$ in rectangular form.

$$\begin{align*}r \sin \theta & = -3 \\ y & = -3\end{align*}$$

Example 4

Write the polar equation $\begin{align*}r = 2 \sin \theta\end{align*}$ in rectangular form.

$$\begin{align*}r & = 2 \sin \theta \\ r^2 & = 2 r \sin \theta \\ x^2 + y^2 & = 2 y \\ y^2 - 2y & = - x^2 \\ y^2 - 2y + 1 & = -x^2 + 1 \\ (y - 1)^2 & = -x^2 +1 \\ x^2 + (y - 1)^2 & = 1\end{align*}$$

Review

Given the following polar coordinates, find the corresponding rectangular coordinates of the points.

1. $\begin{align*}(2, \frac{\pi}{6})\end{align*}$
2. $\begin{align*}(4, \frac{2\pi}{3})\end{align*}$
3. $\begin{align*}(5, \frac{\pi}{3})\end{align*}$
4. $\begin{align*}(3, \frac{\pi}{4})\end{align*}$
5. $\begin{align*}(6, \frac{3\pi}{4})\end{align*}$

Write each polar equation in rectangular form.

6. $\begin{align*}r=3\sin \theta \end{align*}$
7. $\begin{align*}r=2\cos \theta \end{align*}$
8. $\begin{align*}r=5\csc \theta \end{align*}$
9. $\begin{align*}r=3\sec \theta \end{align*}$
10. $\begin{align*}r=6\cos \theta \end{align*}$
11. $\begin{align*}r=8\sin \theta \end{align*}$
12. $\begin{align*}r=2\csc \theta \end{align*}$
13. $\begin{align*}r=4\sec \theta \end{align*}$
14. $\begin{align*}r=3\cos \theta \end{align*}$
15. $\begin{align*}r=5\sin \theta \end{align*}$

Review (Answers)

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