Convert to Polar y=-( square root of 3)/3x

$y = - \frac{\sqrt{3}}{3} x$

Step 1

Since $x = r \cos (θ)$ , replace $x$ with $r \cos (θ)$ .

$y = - \frac{\sqrt{3}}{3} \cdot (r \cos (θ))$

Step 2

Since $y = r \sin (θ)$ , replace $y$ with $r \sin (θ)$ .

$r \sin (θ) = - \frac{\sqrt{3}}{3} \cdot (r \cos (θ))$

Step 3

Solve for $r$ .

Simplify the right side.

Simplify $- \frac{\sqrt{3}}{3} \cdot (r \cos (θ))$ .

Multiply $- \frac{\sqrt{3}}{3} (r \cos (θ))$ .

Combine $r$ and $\frac{\sqrt{3}}{3}$ .

$r \sin (θ) = - \frac{r \sqrt{3}}{3} \cos (θ)$

Combine $\cos (θ)$ and $\frac{r \sqrt{3}}{3}$ .

$r \sin (θ) = - \frac{\cos (θ) (r \sqrt{3})}{3}$

Reorder factors in $- \frac{\cos (θ) r \sqrt{3}}{3}$ .

$r \sin (θ) = - \frac{r \cos (θ) \sqrt{3}}{3}$

Multiply both sides by $3$ .

$r \sin (θ) \cdot 3 = - \frac{r \cos (θ) \sqrt{3}}{3} \cdot 3$

Simplify.

Simplify the left side.

Move $3$ to the left of $r \sin (θ)$ .

$3 r \sin (θ) = - \frac{r \cos (θ) \sqrt{3}}{3} \cdot 3$

Simplify the right side.

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{r \cos (θ) \sqrt{3}}{3}$ into the numerator.

$3 r \sin (θ) = \frac{- r \cos (θ) \sqrt{3}}{3} \cdot 3$

Cancel the common factor.

$3 r \sin (θ) = \frac{- r \cos (θ) \sqrt{3}}{3} \cdot 3$

Rewrite the expression.

$3 r \sin (θ) = - r \cos (θ) \sqrt{3}$

Solve for $r$ .

Add $r \cos (θ) \sqrt{3}$ to both sides of the equation.

$3 r \sin (θ) + r \cos (θ) \sqrt{3} = 0$

Factor $r$ out of $3 r \sin (θ) + r \cos (θ) \sqrt{3}$ .

Factor $r$ out of $3 r \sin (θ)$ .

$r (3 \sin (θ)) + r \cos (θ) \sqrt{3} = 0$

Factor $r$ out of $r \cos (θ) \sqrt{3}$ .

$r (3 \sin (θ)) + r (\cos (θ) \sqrt{3}) = 0$

Factor $r$ out of $r (3 \sin (θ)) + r (\cos (θ) \sqrt{3})$ .

$r (3 \sin (θ) + \cos (θ) \sqrt{3}) = 0$

Divide each term in $r (3 \sin (θ) + \cos (θ) \sqrt{3}) = 0$ by $3 \sin (θ) + \cos (θ) \sqrt{3}$ and simplify.

Divide each term in $r (3 \sin (θ) + \cos (θ) \sqrt{3}) = 0$ by $3 \sin (θ) + \cos (θ) \sqrt{3}$ .

$\frac{r (3 \sin (θ) + \cos (θ) \sqrt{3})}{3 \sin (θ) + \cos (θ) \sqrt{3}} = \frac{0}{3 \sin (θ) + \cos (θ) \sqrt{3}}$

Simplify the left side.

Cancel the common factor of $3 \sin (θ) + \cos (θ) \sqrt{3}$ .

Cancel the common factor.

$\frac{r (3 \sin (θ) + \cos (θ) \sqrt{3})}{3 \sin (θ) + \cos (θ) \sqrt{3}} = \frac{0}{3 \sin (θ) + \cos (θ) \sqrt{3}}$

Divide $r$ by $1$ .

$r = \frac{0}{3 \sin (θ) + \cos (θ) \sqrt{3}}$

Simplify the right side.

Divide $0$ by $3 \sin (θ) + \cos (θ) \sqrt{3}$ .

$r = 0$

Convert to Polar y=-( square root of 3)/3x

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