# Polar Equation: Conversion Between Rectangular Form

When converting between polar coordinates and rectangular coordinates it is much straightforward to convert from polar coordinates to rectangular coordinates. However the conversion from rectangular coordinates to polar coordinates requires more work. When converting equations it is more complicated to convert from polar to rectangular form.

To change a rectangular equation to a polar equation just replace x with $r\mathrm{cos}\theta$ and y with $r\mathrm{sin}\theta$.

RECTANGULAR-POLAR EQUATION CONVERSION:

Substitute the following for x and y:

$x=r\mathrm{cos}\theta$$y=r\mathrm{sin}\theta$

To change a polar equation to a rectangular equation is more difficult and hence we will explore just the simplest of polar equations where the polar equation contains r or θ but not both. The following relationships will be used:

POLAR - RECTANGULAR EQUATION CONVERSION:

Substitute the following for r and θ:

$r=\sqrt{{x}^{2}+{y}^{2}}$$\theta =\frac{y}{x}$

Let's convert between rectangular and polar equations in some examples:

Example 1: Convert the rectangular equations to polar form:
a)   ${x}^{2}+{y}^{2}=16$     b)   x = 6
 Step 1: Substitute for x and y in x2 + y2 = 16 and solve for r. $x=r\mathrm{cos}\theta$, $y=r\mathrm{sin}\theta$ x2 + y2 = 16 ${\left(r\mathrm{cos}\theta \right)}^{2}+{\left(r\mathrm{sin}\theta \right)}^{2}=16$ Sub Solve for r ${r}^{2}{\mathrm{cos}}^{2}\theta +{r}^{2}{\mathrm{sin}}^{2}\theta =16$ Square ${r}^{2}\left({\mathrm{cos}}^{2}\theta +{\mathrm{sin}}^{2}\theta \right)=16$ Factor r2 r2(1) = 16 Trig id. r2 = 16      Multiply r = 4      Take root Step 2: Substitute for x and y in x = 6 and solve for r. $x=r\mathrm{cos}\theta$, $y=r\mathrm{sin}\theta$ x = 6 $r\mathrm{cos}\theta =6$ Sub Solve for r $r=\frac{6}{\mathrm{cos}\theta }$     $÷$ by $\mathrm{cos}\theta$ $r=6\mathrm{sec}\theta$ Simplify
Example 2: Convert the following polar equation to rectangular equations.
a)   r = 5     b)   $\theta =\pi /6$
 Step 1: Square both sides of r = 5 and substitute for r2. ${r}^{2}={x}^{2}+{y}^{2}$ r = 5 ${r}^{2}={5}^{2}=25$    Square ${x}^{2}+{y}^{2}=25$    Sub Step 2: Determine the value of and equate this to $\frac{y}{x}$. $\mathrm{tan}\theta =\frac{y}{x}$ $\theta =\frac{\pi }{6}$ $\mathrm{tan}\frac{\pi }{6}=\frac{\sqrt{3}}{3}$    Find tan θ $\frac{\sqrt{3}}{3}=\frac{y}{x}$        Sub $y=\frac{\sqrt{3}x}{3}$

 Related Links: Math algebra Conversion from Polar to Rectangular Form Complex Parametric Equations: Introduction Pre Calculus

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