Rewrite the Cartesian Equation as a Polar Equation x^2+(y-2)^2=4

$x^{2} + {(y - 2)}^{2} = 4$

Step 1

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

${(r \cos (θ))}^{2} + {(y - 2)}^{2} = 4$

Step 2

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

${(r \cos (θ))}^{2} + {((r \sin (θ)) - 2)}^{2} = 4$

Step 3

Solve for $r$ .

Subtract $4$ from both sides of the equation.

${(r \cos (θ))}^{2} + {((r \sin (θ)) - 2)}^{2} - 4 = 0$

Simplify the left side of the equation.

Simplify each term.

Apply the product rule to $r \cos (θ)$ .

$r^{2} \cos^{2} (θ) + {((r \sin (θ)) - 2)}^{2} - 4 = 0$

Rewrite ${(r \sin (θ) - 2)}^{2}$ as $(r \sin (θ) - 2) (r \sin (θ) - 2)$ .

$r^{2} \cos^{2} (θ) + (r \sin (θ) - 2) (r \sin (θ) - 2) - 4 = 0$

Expand $(r \sin (θ) - 2) (r \sin (θ) - 2)$ using the FOIL Method.

Apply the distributive property.

$r^{2} \cos^{2} (θ) + r \sin (θ) (r \sin (θ) - 2) - 2 (r \sin (θ) - 2) - 4 = 0$

Apply the distributive property.

$r^{2} \cos^{2} (θ) + r \sin (θ) (r \sin (θ)) + r \sin (θ) \cdot - 2 - 2 (r \sin (θ) - 2) - 4 = 0$

Apply the distributive property.

$r^{2} \cos^{2} (θ) + r \sin (θ) (r \sin (θ)) + r \sin (θ) \cdot - 2 - 2 (r \sin (θ)) - 2 \cdot - 2 - 4 = 0$

Simplify and combine like terms.

Simplify each term.

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

Move $r$ .

$r^{2} \cos^{2} (θ) + r \cdot r \sin (θ) \sin (θ) + r \sin (θ) \cdot - 2 - 2 (r \sin (θ)) - 2 \cdot - 2 - 4 = 0$

Multiply $r$ by $r$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin (θ) \sin (θ) + r \sin (θ) \cdot - 2 - 2 (r \sin (θ)) - 2 \cdot - 2 - 4 = 0$

Multiply $r^{2} \sin (θ) \sin (θ)$ .

Raise $\sin (θ)$ to the power of $1$ .

$r^{2} \cos^{2} (θ) + r^{2} (\sin (θ) \sin (θ)) + r \sin (θ) \cdot - 2 - 2 (r \sin (θ)) - 2 \cdot - 2 - 4 = 0$

Raise $\sin (θ)$ to the power of $1$ .

$r^{2} \cos^{2} (θ) + r^{2} (\sin (θ) \sin (θ)) + r \sin (θ) \cdot - 2 - 2 (r \sin (θ)) - 2 \cdot - 2 - 4 = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r^{2} \cos^{2} (θ) + r^{2} {\sin (θ)}^{1 + 1} + r \sin (θ) \cdot - 2 - 2 (r \sin (θ)) - 2 \cdot - 2 - 4 = 0$

Add $1$ and $1$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) + r \sin (θ) \cdot - 2 - 2 (r \sin (θ)) - 2 \cdot - 2 - 4 = 0$

Move $- 2$ to the left of $r \sin (θ)$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 2 \cdot (r \sin (θ)) - 2 (r \sin (θ)) - 2 \cdot - 2 - 4 = 0$

Multiply $- 2$ by $- 2$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 2 r \sin (θ) - 2 r \sin (θ) + 4 - 4 = 0$

Subtract $2 r \sin (θ)$ from $- 2 r \sin (θ)$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 4 r \sin (θ) + 4 - 4 = 0$

Simplify terms.

Combine the opposite terms in $r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 4 r \sin (θ) + 4 - 4$ .

Subtract $4$ from $4$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 4 r \sin (θ) + 0 = 0$

Add $r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 4 r \sin (θ)$ and $0$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 4 r \sin (θ) = 0$

Factor $r^{2}$ out of $r^{2} \cos^{2} (θ)$ .

$r^{2} (\cos^{2} (θ)) + r^{2} \sin^{2} (θ) - 4 r \sin (θ) = 0$

Factor $r^{2}$ out of $r^{2} \sin^{2} (θ)$ .

$r^{2} (\cos^{2} (θ)) + r^{2} (\sin^{2} (θ)) - 4 r \sin (θ) = 0$

Factor $r^{2}$ out of $r^{2} (\cos^{2} (θ)) + r^{2} (\sin^{2} (θ))$ .

$r^{2} (\cos^{2} (θ) + \sin^{2} (θ)) - 4 r \sin (θ) = 0$

Rearrange terms.

$r^{2} (\sin^{2} (θ) + \cos^{2} (θ)) - 4 r \sin (θ) = 0$

Apply pythagorean identity.

$r^{2} \cdot 1 - 4 r \sin (θ) = 0$

Multiply $r^{2}$ by $1$ .

$r^{2} - 4 r \sin (θ) = 0$

Factor $r$ out of $r^{2} - 4 r \sin (θ)$ .

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

$r \cdot r - 4 r \sin (θ) = 0$

Factor $r$ out of $- 4 r \sin (θ)$ .

$r \cdot r + r (- 4 \sin (θ)) = 0$

Factor $r$ out of $r \cdot r + r (- 4 \sin (θ))$ .

$r (r - 4 \sin (θ)) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$r = 0$

$r - 4 \sin (θ) = 0$

Set $r$ equal to $0$ .

$r = 0$

Set $r - 4 \sin (θ)$ equal to $0$ and solve for $r$ .

Set $r - 4 \sin (θ)$ equal to $0$ .

$r - 4 \sin (θ) = 0$

Add $4 \sin (θ)$ to both sides of the equation.

$r = 4 \sin (θ)$

The final solution is all the values that make $r (r - 4 \sin (θ)) = 0$ true.

$r = 0$

$r = 4 \sin (θ)$

$r = 0$

$r = 4 \sin (θ)$

Rewrite the Cartesian Equation as a Polar Equation x^2+(y-2)^2=4

Related Questions

Support LCMGCF.com by disabling your adblocker.