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:

$x = r \cos θ$ $y = r \sin θ$

$r^{2} = x^{2} + y^{2}$ $\tan θ = \frac{y}{x}$

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

Take the polar equation

r = \sec θ

. The variable r can be substituted by

\sqrt{x^{2} + y^{2}}

but there is no direct substitution for

\sec θ

. 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

\sec θ = \frac{1}{\cos θ}

, rewrite the equation as

r = \frac{1}{\cos θ}

.

In this form, note that if

\frac{1}{\cos θ}

is multiplied by

\frac{1}{r}

to give

\frac{1}{r \cos θ}

, then

r \cos θ

can be replaced by x. To keep our equation balance, let's multiplied both sides by

\frac{1}{r}

$r \cdot \frac{1}{r} = \frac{1}{\cos θ} \cdot \frac{1}{r} \to 1 = \frac{1}{r \cos θ}$

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.

$\sec θ = \frac{1}{\cos θ}$

Original

$r = 6 \sec θ$

Rewrite sec θ in terms of cos θ.

$\sec θ = \frac{1}{\cos θ}$

Trigonometric substitution into original

$r = 6 \frac{1}{\cos θ}$

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 $x = r \cos θ$ .

$r = 6 \frac{1}{\cos θ}$

Applicable Polar-Rectangular relationship

$x = r \cos θ$

Multiple both sides by $\frac{1}{r}$ to obtain $\frac{1}{r \cos θ}$

$\frac{r}{r} = 6 \frac{1}{r \cos θ}$

Step 3: Substitute and simplify the equation.

Substitute for r cos θ

$1 = 6 (\frac{1}{x})$

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

$r^{2} = 3 \sin 2 θ$

The term sin 2θ is a double angle and needs to be replaced by the double angle identity, sin 2θ = 2sin θ cos θ.

$r^{2} = 3 \sin 2 θ$

$r^{2} = 3 (2 \sin θ \cos θ)$

$r^{2} = 6 \sin θ \cos θ$

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.

$r^{2} \cdot r^{2} = 6 \sin θ \cos θ \cdot r^{2}$

${(r^{2})}^{2} = 6 r \sin θ r \cos θ$

Step 3: Substitute and simplify the equation.

r² = x² + y²

x = r cos θ

y = r sin θ

Make the polar-rectangular substitutions.

${(x^{2} + y^{2})}^{2} = 6 y x$

Simplify

$x^{4} + 2 x^{2} y^{2} + y^{4} = 6 x y$

$x^{4} + 2 x^{2} y^{2} + y^{4} - 6 x = 0$

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Conversion from Polar to Rectangular Form Complex

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