Conversion from Polar to Rectangular Form Complex
RECTANGULAR-POLAR RELATIONSHIPS:
$x=r\mathrm{cos}\theta $$y=r\mathrm{sin}\theta $
${r}^{2}={x}^{2}+{y}^{2}$$\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 $\frac{1}{r}$.
$r\xb7\frac{1}{r}=\frac{1}{\mathrm{cos}\theta}\xb7\frac{1}{r}\to 1=\frac{1}{r\mathrm{cos}\theta}$
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.
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 $r=6\mathrm{sec}\theta $ 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 |
Step 1: Rewrite any trigonometric function in terms of cos, sin, or tan. |
Original ${r}^{2}=3\mathrm{sin}2\theta $ 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\mathrm{sin}2\theta $ ${r}^{2}=3\left(2\mathrm{sin}\theta \mathrm{cos}\theta \right)$ ${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 r^{2} = x^{2} + y^{2}. |
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 r^{2}. ${r}^{2}\xb7{r}^{2}=6\mathrm{sin}\theta \mathrm{cos}\theta \xb7{r}^{2}$ ${\left({r}^{2}\right)}^{2}=6r\mathrm{sin}\theta r\mathrm{cos}\theta $ |
Step 3: Substitute and simplify the equation. r^{2} = x^{2} + y^{2} 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 |
To link to this Conversion from Polar to Rectangular Form Complex page, copy the following code to your site: