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# 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 $x$ and $y$ are usually used to designate the rectangular coordinates of a point, $r$ and $\theta$ are usually used to designate the polar coordinates of the point. $r$ is the distance of the point to the origin. $\theta$ is the angle that the line from the origin to the point makes with the positive $x-$ axis. The diagram below shows both polar and Cartesian coordinates applied to a point $P$ . By applying trigonometry, we can obtain equations that will show the relationship between polar coordinates $(r, \theta)$ and the rectangular coordinates $(x, y)$

The point $P$ has the polar coordinates $(r, \theta)$ and the rectangular coordinates $(x, y)$ .

Therefore

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

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: $W(4,-200^\circ),H \left (4, \frac{\pi}{3} \right )$

Solution:

a) For $W(4,-200^\circ), r = 4$ and $\theta = -200^\circ$

$x & = r \cos \theta && y = r \sin \theta\\x &= 4 \cos (-200^\circ) && y = 4 \sin(-200^\circ)\\x &= 4(-.9396) && y = 4(.3420)\\x & \approx - 3.76 && y \approx 1.37$

The rectangular coordinates of $W$ are approximately $(-3.76, 1.37)$ .

b) For $H \left ( 4,\frac{\pi}{3} \right ), r = 4$ and $\theta = \frac{\pi}{3}$

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

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

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 $r = 4 \cos \theta$ in rectangular form.

Solution:

$r &= 4 \cos \theta\\r^2 &= 4r \cos \theta && Multiply \ both \ sides \ by \ r.\\ x^2 + y^2 &= 4x && r^2 = x^2 + y^2 \ and \ x = r \cos \theta$

The equation is now in rectangular form. The $r^2$ and $\theta$ 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.

$x^2 - 4x + y^2 &= 0\\x^2 - 4x + 4 + y^2 &= 4 && Complete \ the \ square \ for \ x^2 - 4x.\\ (x - 2)^2 + y^2 &= 4 && Factor \ x^2 - 4x + 4.$

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 $r = 4 \cos \theta$ for $0 \le \theta \le 2 \pi$ or the rectangular form $(x - 2)^2 + y^2 = 4.$

#### Example C

Write the polar equation $r = 3 \csc \theta$ in rectangular form.

Solution:

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

### 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 $r = 6 \cos \theta$ in rectangular form.

2. Write the polar equation $r \sin \theta = -3$ in rectangular form.

3. Write the polar equation $r = 2 \sin \theta$ in rectangular form.

Solutions:

1.

$r & = 6 \cos \theta \\r^2 & = 6r \cos \theta \\x^2 + y^2 & = 6x \\x^2 - 6x + y^2 & = 0 \\x^2 - 6x + 9 + y^2 & = 9 \\(x - 3)^2 + y^2 & = 9$

2.

$r \sin \theta & = -3 \\y & = -3$

3.

$r & = 2 \sin \theta \\r^2 & = 2 r \sin \theta \\x^2 + y^2 & = 2 y \\y^2 - 2y & = - x^2 \\y^2 - 2y + 1 & = -x^2 + 1 \\(y - 1)^2 & = -x^2 +1 \\x^2 + (y - 1)^2 & = 1$

### Concept Problem Solution

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

$x & = r \cos \theta && y = r \sin \theta\\x &= 3 \cos (70^\circ) && y = 3 \sin(70^\circ)\\x &= 3(.342) && y = 3(.94)\\x & \approx 1.026 && y \approx 2.82$

### Practice

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

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

Write each polar equation in rectangular form.

1. $r=3\sin \theta$
2. $r=2\cos \theta$
3. $r=5\csc \theta$
4. $r=3\sec \theta$
5. $r=6\cos \theta$
6. $r=8\sin \theta$
7. $r=2\csc \theta$
8. $r=4\sec \theta$
9. $r=3\cos \theta$
10. $r=5\sin \theta$

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