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# 6.4: Polar to Rectangular Conversions

Difficulty Level: At Grade Created by: CK-12
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Practice Polar to Rectangular Conversions

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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?

### Guidance

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.

#### Example A

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*}

Solution:

a) 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*}.

b) 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*}.

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

#### Example B

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

Solution:

\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*}

#### Example C

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

Solution:

\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*}

### Vocabulary

Polar Coordinates: A set of polar coordinates are a set of coordinates plotted on a system that uses the distance from the origin and angle from an axis to describe location.

Rectangular Coordinates: A set of rectangular coordinates are a set of coordinates plotted on a system using basis axes at right angles to each other.

### Guided Practice

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

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

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

Solutions:

1.

\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*}

2.

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

3.

\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*}

### Concept Problem Solution

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*}

### Practice

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.

1. \begin{align*}r=3\sin \theta \end{align*}
2. \begin{align*}r=2\cos \theta \end{align*}
3. \begin{align*}r=5\csc \theta \end{align*}
4. \begin{align*}r=3\sec \theta \end{align*}
5. \begin{align*}r=6\cos \theta \end{align*}
6. \begin{align*}r=8\sin \theta \end{align*}
7. \begin{align*}r=2\csc \theta \end{align*}
8. \begin{align*}r=4\sec \theta \end{align*}
9. \begin{align*}r=3\cos \theta \end{align*}
10. \begin{align*}r=5\sin \theta \end{align*}

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