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Polar to Rectangular Conversions

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Polar to 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?

Watch This

James Sousa Example: Convert a point in polar coordinates to rectangular coordinates

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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