Rewrite the Cartesian Equation as a Polar Equation y=x^2

$y = x^{2}$

Step 1

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

$y = {(r \cos (θ))}^{2}$

Step 2

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

$r \sin (θ) = {(r \cos (θ))}^{2}$

Step 3

Solve for $r$ .

Subtract ${(r \cos (θ))}^{2}$ from both sides of the equation.

$r \sin (θ) - {(r \cos (θ))}^{2} = 0$

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

$r \sin (θ) - r^{2} \cos^{2} (θ) = 0$

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

Factor $r$ out of $r \sin (θ)$ .

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

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

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

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

$r (\sin (θ) - r \cos^{2} (θ)) = 0$

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

$r = 0$

$\sin (θ) - r \cos^{2} (θ) = 0$

Set $r$ equal to $0$ .

$r = 0$

Set $\sin (θ) - r \cos^{2} (θ)$ equal to $0$ and solve for $r$ .

Set $\sin (θ) - r \cos^{2} (θ)$ equal to $0$ .

$\sin (θ) - r \cos^{2} (θ) = 0$

Solve $\sin (θ) - r \cos^{2} (θ) = 0$ for $r$ .

Subtract $\sin (θ)$ from both sides of the equation.

$- r \cos^{2} (θ) = - \sin (θ)$

Divide each term in $- r \cos^{2} (θ) = - \sin (θ)$ by $- 1 \cos^{2} (θ)$ and simplify.

Divide each term in $- r \cos^{2} (θ) = - \sin (θ)$ by $- 1 \cos^{2} (θ)$ .

$\frac{- r \cos^{2} (θ)}{- 1 \cos^{2} (θ)} = \frac{- \sin (θ)}{- 1 \cos^{2} (θ)}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{r \cos^{2} (θ)}{\cos^{2} (θ)} = \frac{- \sin (θ)}{- 1 \cos^{2} (θ)}$

Cancel the common factor of $\cos^{2} (θ)$ .

Cancel the common factor.

$\frac{r \cos^{2} (θ)}{\cos^{2} (θ)} = \frac{- \sin (θ)}{- 1 \cos^{2} (θ)}$

Divide $r$ by $1$ .

$r = \frac{- \sin (θ)}{- 1 \cos^{2} (θ)}$

Simplify the right side.

Dividing two negative values results in a positive value.

$r = \frac{\sin (θ)}{\cos^{2} (θ)}$

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

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

Separate fractions.

$r = \frac{1}{\cos (θ)} \cdot \frac{\sin (θ)}{\cos (θ)}$

Convert from $\frac{\sin (θ)}{\cos (θ)}$ to $\tan (θ)$ .

$r = \frac{1}{\cos (θ)} \tan (θ)$

Convert from $\frac{1}{\cos (θ)}$ to $\sec (θ)$ .

$r = \sec (θ) \tan (θ)$

The final solution is all the values that make $r (\sin (θ) - r \cos^{2} (θ)) = 0$ true.

$r = 0$

$r = \sec (θ) \tan (θ)$

$r = 0$

$r = \sec (θ) \tan (θ)$

Rewrite the Cartesian Equation as a Polar Equation y=x^2

Related Questions

Support LCMGCF.com by disabling your adblocker.