Convert to Rectangular rsin(theta)=-2

$r \sin (θ) = - 2$

Step 1

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

$r \frac{y}{r} = - 2$

Step 2

Multiply both sides by $r$ .

$r (r \frac{y}{r}) = r \cdot - 2$

Simplify the left side.

Simplify $r (r \frac{y}{r})$ .

Multiply $r$ by $r$ by adding the exponents.

Move $r$ .

$r \cdot r \frac{y}{r} = r \cdot - 2$

Multiply $r$ by $r$ .

$r^{2} \frac{y}{r} = r \cdot - 2$

Cancel the common factor of $r$ .

Factor $r$ out of $r^{2}$ .

$r \cdot r \frac{y}{r} = r \cdot - 2$

Cancel the common factor.

$r \cdot r \frac{y}{r} = r \cdot - 2$

Rewrite the expression.

$r y = r \cdot - 2$

Simplify the right side.

Move $- 2$ to the left of $r$ .

$r y = - 2 r$

Step 3

Since $r^{2} = x^{2} + y^{2}$ , replace $r^{2}$ with $x^{2} + y^{2}$ and $r$ with $\sqrt{x^{2} + y^{2}}$ .

$\sqrt{x^{2} + y^{2}} y = - 2 \sqrt{x^{2} + y^{2}}$

Step 4

Solve for $y$ .

Solve for $\sqrt{x^{2} + y^{2}}$ .

Add $2 \sqrt{x^{2} + y^{2}}$ to both sides of the equation.

$\sqrt{x^{2} + y^{2}} y + 2 \sqrt{x^{2} + y^{2}} = 0$

Factor $\sqrt{x^{2} + y^{2}}$ out of $\sqrt{x^{2} + y^{2}} y + 2 \sqrt{x^{2} + y^{2}}$ .

Factor $\sqrt{x^{2} + y^{2}}$ out of $2 \sqrt{x^{2} + y^{2}}$ .

$\sqrt{x^{2} + y^{2}} y + \sqrt{x^{2} + y^{2}} \cdot 2 = 0$

Factor $\sqrt{x^{2} + y^{2}}$ out of $\sqrt{x^{2} + y^{2}} y + \sqrt{x^{2} + y^{2}} \cdot 2$ .

$\sqrt{x^{2} + y^{2}} (y + 2) = 0$

Divide each term in $\sqrt{x^{2} + y^{2}} (y + 2) = 0$ by $y + 2$ and simplify.

Divide each term in $\sqrt{x^{2} + y^{2}} (y + 2) = 0$ by $y + 2$ .

$\frac{\sqrt{x^{2} + y^{2}} (y + 2)}{y + 2} = \frac{0}{y + 2}$

Simplify the left side.

Cancel the common factor of $y + 2$ .

Cancel the common factor.

$\frac{\sqrt{x^{2} + y^{2}} (y + 2)}{y + 2} = \frac{0}{y + 2}$

Divide $\sqrt{x^{2} + y^{2}}$ by $1$ .

$\sqrt{x^{2} + y^{2}} = \frac{0}{y + 2}$

Simplify the right side.

Divide $0$ by $y + 2$ .

$\sqrt{x^{2} + y^{2}} = 0$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{2} + y^{2}}}^{2} = 0^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + y^{2}}$ as ${(x^{2} + y^{2})}^{\frac{1}{2}}$ .

${({(x^{2} + y^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Simplify the left side.

Simplify ${({(x^{2} + y^{2})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x^{2} + y^{2})}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} + y^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} + y^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

${(x^{2} + y^{2})}^{1} = 0^{2}$

Simplify.

$x^{2} + y^{2} = 0^{2}$

Simplify the right side.

Raising $0$ to any positive power yields $0$ .

$x^{2} + y^{2} = 0$

Solve for $y$ .

Subtract $x^{2}$ from both sides of the equation.

$y^{2} = - x^{2}$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- x^{2}}$

Simplify $\pm \sqrt{- x^{2}}$ .

Reorder $- 1$ and $x^{2}$ .

$y = \pm \sqrt{x^{2} \cdot - 1}$

Pull terms out from under the radical.

$y = \pm x \sqrt{- 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \pm x i$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$y = x i$

Next, use the negative value of the $\pm$ to find the second solution.

$y = - x i$

The complete solution is the result of both the positive and negative portions of the solution.

$y = x i$

$y = - x i$

$y = x i$

$y = - x i$

$y = x i$

$y = - x i$

$y = x i$

$y = - x i$

Step 5

Convert to Rectangular rsin(theta)=-2

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