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

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

Therefore

xy=rcosθ=rsinθr2=x2+y2tanθ=yx\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: W(4,200),H(4,π3)\begin{align*}W(4,-200^\circ),H \left (4, \frac{\pi}{3} \right )\end{align*}

Solution:

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

xxxx=rcosθ=4cos(200)=4(.9396)3.76y=rsinθy=4sin(200)y=4(.3420)y1.37\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 W\begin{align*}W\end{align*} are approximately (3.76,1.37)\begin{align*}(-3.76, 1.37)\end{align*}.

b) For H(4,π3),r=4\begin{align*}H \left ( 4,\frac{\pi}{3} \right ), r = 4\end{align*} and θ=π3\begin{align*}\theta = \frac{\pi}{3}\end{align*}

xxxx=rcosθ=4cosπ3=4(12)=2y=rsinθy=4sinπ3y=4(32)y=23\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 H\begin{align*}H\end{align*} are (2,23)\begin{align*}(2, 2 \sqrt{3})\end{align*} or approximately (2,3.46)\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 r=4cosθ\begin{align*}r = 4 \cos \theta \end{align*} in rectangular form.

Solution:

rr2x2+y2=4cosθ=4rcosθ=4xMultiply both sides by r.r2=x2+y2 and x=rcosθ\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 r2\begin{align*}r^2\end{align*} and θ\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.

x24x+y2x24x+4+y2(x2)2+y2=0=4=4Complete the square for x24x.Factor x24x+4.\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 r=4cosθ\begin{align*}r = 4 \cos \theta\end{align*} for 0θ2π\begin{align*}0 \le \theta \le 2 \pi\end{align*} or the rectangular form (x2)2+y2=4.\begin{align*}(x - 2)^2 + y^2 = 4.\end{align*}

Example C

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

Solution:

rrcscθr1cscθrsinθy=3cscθ=3=3=3=3divide bycscθsinθ=1cscθy=rsinθ\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 r=6cosθ\begin{align*}r = 6 \cos \theta\end{align*} in rectangular form.

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

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

Solutions:

1.

rr2x2+y2x26x+y2x26x+9+y2(x3)2+y2=6cosθ=6rcosθ=6x=0=9=9\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.

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

3.

rr2x2+y2y22yy22y+1(y1)2x2+(y1)2=2sinθ=2rsinθ=2y=x2=x2+1=x2+1=1\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 (3,70)\begin{align*}(3,70^\circ)\end{align*}. Therefore, the radius is equal to 3 and the angle is equal to 70\begin{align*}70^\circ\end{align*}. The rectangular coordinates of this point can be found as follows:

xxxx=rcosθ=3cos(70)=3(.342)1.026y=rsinθy=3sin(70)y=3(.94)y2.82\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. (2,π6)\begin{align*}(2, \frac{\pi}{6})\end{align*}
2. (4,2π3)\begin{align*}(4, \frac{2\pi}{3})\end{align*}
3. (5,π3)\begin{align*}(5, \frac{\pi}{3})\end{align*}
4. (3,π4)\begin{align*}(3, \frac{\pi}{4})\end{align*}
5. (6,3π4)\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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