# Conversion from Polar to Rectangular Form Complex

If a polar equation is written such that it contains terms that appear in the polar-rectangular relationships (see below), conversion from a polar equation to a rectangular equation is a simple matter of substitution.

RECTANGULAR-POLAR RELATIONSHIPS:

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

$\mathrm{tan}\theta =\frac{y}{x}$

In situations where this is not the case, the process of conversion requires some creative manipulation.

Take the polar equation $r=\mathrm{sec}\theta$. The variable r can be substituted by $\sqrt{{x}^{2}+{y}^{2}}$ but there is no direct substitution for $\mathrm{sec}\theta$. This is where the creative manipulation comes in.

Note that the given rectangular-polar relationships involve cos θ, sin θ and tan θ. Thus the first step is to rewrite sec θ in terms of cos, sin, or tan. Since $\mathrm{sec}\theta =\frac{1}{\mathrm{cos}\theta }$, rewrite the equation as $r=\frac{1}{\mathrm{cos}\theta }$.

In this form, note that if $\frac{1}{\mathrm{cos}\theta }$ is multiplied by $\frac{1}{r}$ to give $\frac{1}{r\mathrm{cos}\theta }$, then $r\mathrm{cos}\theta$ can be replaced by x. To keep our equation balance, let's multiplied both sides by $\frac{1}{r}$.

Now r cos θ can be replaced by x resulting in the rectangular equation:

$1=\frac{1}{x}\to x=1$

In general, when converting polar equations to rectangular equations use the following guidelines:

GUIDELINES FOR CONVERTING POLAR TO RECTANGULAR EQUATIONS:

1. Rewrite any trigonometric functions in terms of cos θ, sin θ, or tan θ.

2. Manipulate the equation to obtain terms so that the rectangular-polar substitutions can be used.

3. Substitute and simplify the equation.

Let's try a couple of examples.

Example 1: Convert the polar equation r = 6 sec θ to a rectangular equation.
 Step 1: Rewrite any trigonometric function in terms of cos θ, sin θ, or tan θ. $\mathrm{sec}\theta =\frac{1}{\mathrm{cos}\theta }$ Original Rewrite sec θ in terms of cos θ. $\mathrm{sec}\theta =\frac{1}{\mathrm{cos}\theta }$ Trigonometric substitution into original $r=6\frac{1}{\mathrm{cos}\theta }$ Step 2: Manipulate the equation to obtain terms so that the rectangular-polar substitutions can be used. In this case the equation is manipulated to use the polar-rectangular relationship $x=r\mathrm{cos}\theta$. $r=6\frac{1}{\mathrm{cos}\theta }$ Applicable Polar-Rectangular relationship $x=r\mathrm{cos}\theta$ Multiple both sides by $\frac{1}{r}$ to obtain $\frac{1}{r\mathrm{cos}\theta }$ $\frac{r}{r}=6\frac{1}{r\mathrm{cos}\theta }$ Step 3: Substitute and simplify the equation. Substitute for r cos θ $1=6\left(\frac{1}{x}\right)$ Simplify $1=\frac{6}{x}$ x = 6
Example 2: Convert the polar equation r2 = 3 sin 2θ to a rectangular equation.
 Step 1: Rewrite any trigonometric function in terms of cos, sin, or tan. Original The term sin 2θ is a double angle and needs to be replaced by the double angle identity, sin 2θ = 2sin θ cos θ. ${r}^{2}=3\left(2\mathrm{sin}\theta \mathrm{cos}\theta \right)$ ${r}^{2}=6\mathrm{sin}\theta \mathrm{cos}\theta$ Step 2: Manipulate the equation to obtain terms so that the rectangular-polar substitutions can be used. In this case the equation is manipulated to use the polar-rectangular relationships x = r cos θ, y = r sin θ, and r2 = x2 + y2. To use the polar-rectangular relationships we need r cos θ and r sin θ. To obtain these terms requires each side to be multiplied by r2. ${r}^{2}·{r}^{2}=6\mathrm{sin}\theta \mathrm{cos}\theta ·{r}^{2}$ Step 3: Substitute and simplify the equation. r2 = x2 + y2 x = r cos θ y = r sin θ Make the polar-rectangular substitutions. ${\left({x}^{2}+{y}^{2}\right)}^{2}=6yx$ Simplify ${x}^{4}+2{x}^{2}{y}^{2}+{y}^{4}=6xy$ ${x}^{4}+2{x}^{2}{y}^{2}+{y}^{4}-6x=0$

 Related Links: Math algebra Parametric Equations: Introduction Parametric Equations: Eliminating Parameters Pre Calculus

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