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Rectangular to Polar Form for Equations

Convert equations by substituting x and y with expressions of distance and angles.

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Practice Rectangular to Polar Form for Equations
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Rectangular to Polar Form for Equations

You are working diligently in your math class when your teacher gives you an equation to graph:

$(x + 1)^2 - (y + 2)^2 = 7$

As you start to consider how to rearrange this equation, you are told that the goal of the class is to convert the equation to polar form instead of rectangular form.

Can you find a way to do this?

By the end of this Concept, you'll be able to convert this equation to polar form.

Guidance

Interestingly, a rectangular coordinate system isn't the only way to plot values. A polar system can be useful. However, it will often be the case that there are one or more equations that need to be converted from rectangular to polar form. To write a rectangular equation in polar form, the conversion equations of $x = r \cos \theta$ and $y = r \sin \theta$ are used.

If the graph of the polar equation is the same as the graph of the rectangular equation, then the conversion has been determined correctly.

$(x-2)^2+y^2=4$

The rectangular equation $(x - 2)^2 + y^2 = 4$ represents a circle with center (2, 0) and a radius of 2 units. The polar equation $r = 4 \cos \theta$ is a circle with center (2, 0) and a radius of 2 units.

Example A

Write the rectangular equation $x^2 + y^2 = 2x$ in polar form.

Solution: Remember $r = \sqrt{x^2 + y^2}, r^2 = x^2 + y^2$ and $x = r \cos \theta$ .

$x^2 + y^2 &= 2x\\ r^2 &= 2(r \cos \theta) && Pythagorean \ Theorem \ and \ x = r \cos \theta\\ r^2 &= 2r \cos \theta\\ r &= 2 \cos \theta && Divide \ each \ side \ by \ r$

Example B

Write $(x - 2)^2 + y^2 = 4$ in polar form.

Remember $x = r \cos \theta$ and $y = r \sin \theta$ .

$&(x - 2)^2 + y^2 = 4\\&(r \cos \theta - 2)^2 + (r \sin \theta)^2 = 4 && x = r \cos \theta \ and \ y = r \sin \theta\\ & r^2 \cos^2 \theta - 4r \cos \theta + 4 + r^2 \sin^2 \theta = 4 && expand \ the \ terms\\& r^2 \cos^2 \theta - 4r \cos \theta + r^2 \sin^2 \theta = 0 && subtract \ 4 \ from \ each \ side\\ & r^2 \cos^2 \theta + r^2 \sin^2 \theta = 4r \cos \theta && isolate \ the \ squared \ terms\\ & r^2 (\cos^2 \theta + \sin^2 \theta) = 4r \cos \theta && factor \ r^2 - a \ common \ factor\\ & r^2 = 4r \cos \theta && Pythagorean \ Identity\\ & r = 4 \cos \theta && Divide \ each \ side \ by \ r$

Example C

Write the rectangular equation $(x+4)^2 + (y-1)^2 = 17$ in polar form.

$&(x+4)^2 + (y-1)^2 = 17\\&(r \cos \theta + 4)^2 + (r \sin \theta - 1)^2 = 17 && x = r \cos \theta \ and \ y = r \sin \theta\\ & r^2 \cos^2 \theta + 8r \cos \theta + 16 + r^2 \sin^2 \theta - 2r \sin \theta + 1 = 17 && expand \ the \ terms\\& r^2 \cos^2 \theta + 8r \cos \theta - 2r \sin \theta + r^2 \sin^2 \theta = 0 && subtract \ 17 \ from \ each \ side\\ & r^2 \cos^2 \theta + r^2 \sin^2 \theta = -8r \cos \theta + 2r \sin \theta && isolate \ the \ squared \ terms\\ & r^2 (\cos^2 \theta + \sin^2 \theta) = -2r (4\cos \theta - \sin \theta) && factor \ r^2 - a \ common \ factor\\ & r^2 = -2r (4\cos \theta - \sin \theta) && Pythagorean \ Identity\\ & r = -2(4\cos \theta - \sin \theta) && Divide \ each \ side \ by \ r$

Guided Practice

1. Write the rectangular equation $(x - 4)^2 + (y - 3)^2 = 25$ in polar form.

2. Write the rectangular equation $3x - 2y = 1$ in polar form.

3. Write the rectangular equation $x^2 + y^2 - 4x + 2y = 0$ in polar form.

Solutions:

1.

$(x - 4)^2 + (y - 3)^2 & = 25 \\x^2 - 8x + 16 + y^2 - 6y + 9 & = 25 \\x^2 - 8x + y^2 - 6y + 25 & = 25 \\x^2 - 8x + y^2 - 6y & = 0 \\x^2 + y^2 - 8x - 6y & = 0 \\r^2 - 8(r \cos \theta) - 6(r \sin \theta) & = 0 \\r^2 - 8r \cos \theta - 6r \sin \theta & = 0 \\r(r - 8 \cos \theta - 6 \sin \theta) & = 0 \\r = 0\ \text{or}\ r - 8 \cos \theta - 6 \sin \theta & = 0 \\r = 0\ \text{or}\ r & = 8 \cos \theta + 6 \sin \theta$

From graphing $r-8\cos \theta -6\sin \theta =0$ , we see that the additional solutions are 0 and 8.

2.

$3x - 2y & = 1 \\3r \cos \theta - 2r \sin \theta & = 1 \\r (3 \cos \theta - 2 \sin \theta) & = 1 \\r & = \frac{1}{3 \cos \theta - 2 \sin \theta}$

3.

$x^2 + y^2 - 4x + 2y & = 0 \\r^2 \cos^2 \theta + r^2 \sin^2 \theta - 4 r \cos \theta + 2 r \sin \theta & = 0 \\r^2 (\sin^2 \theta + \cos^2 \theta) - 4 r \cos \theta + 2 r \sin \theta & = 0 \\r (r - 4 \cos \theta + 2 \sin \theta) & = 0 \\r = 0\ \text{or}\ r - 4 \cos \theta + 2 \sin \theta & = 0 \\r = 0\ \text{or}\ r & = 4 \cos \theta - 2 \sin \theta$

Concept Problem Solution

The original equation to convert is:

$(x + 1)^2 - (y + 2)^2 = 7$

You can substitute $x = r\cos \theta$ and $y = r\sin \theta$ into the equation, and then simplify:

$(r\cos \theta +1)^2 - (r\sin \theta + 2)^2 = 7\\(r^2\cos^2 \theta + 2r\cos \theta +1) - (r^2\sin^2 \theta + 4r\sin \theta + 4) = 7\\r^2(\cos^2 \theta - \sin^2 \theta) + 2r(\cos \theta - 2\sin \theta) - 3 = 7\\r^2(\cos^2 \theta - \sin^2 \theta) + 2r(\cos \theta - 2\sin \theta) = 10$

Explore More

Write each rectangular equation in polar form.

1. $x=3$
2. $y=4$
3. $x^2+y^2=4$
4. $x^2+y^2=9$
5. $(x-1)^2+y^2=1$
6. $(x-2)^2+(y-3)^2=13$
7. $(x-1)^2+(y-3)^2=10$
8. $(x+2)^2+(y+2)^2=8$
9. $(x+5)^2+(y-1)^2=26$
10. $x^2+(y-6)^2=36$
11. $x^2+(y+2)^2=4$
12. $2x+5y=11$
13. $4x-7y=10$
14. $x+5y=8$
15. $3x-4y=15$

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