Algebra Examples

$y = 2 x^{2}$

Step 1

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

$y = 2 {(r \cos (θ))}^{2}$

Step 2

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

$r \sin (θ) = 2 {(r \cos (θ))}^{2}$

Step 3

Solve for $r$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

Tap for more steps...

Step 3.3.1

Factor $r$ out of $r \sin (θ)$ .

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

Step 3.3.2

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

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

Step 3.3.3

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

$r (\sin (θ) - 2 r \cos^{2} (θ)) = 0$

Step 3.4

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

$r = 0$

$\sin (θ) - 2 r \cos^{2} (θ) = 0$

Step 3.5

Set $r$ equal to $0$ .

$r = 0$

Step 3.6

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

Tap for more steps...

Step 3.6.1

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

$\sin (θ) - 2 r \cos^{2} (θ) = 0$

Step 3.6.2

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

Tap for more steps...

Step 3.6.2.1

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

$- 2 r \cos^{2} (θ) = - \sin (θ)$

Step 3.6.2.2

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

Tap for more steps...

Step 3.6.2.2.1

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

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

Step 3.6.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.6.2.2.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 3.6.2.2.2.1.1

Cancel the common factor.

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

Step 3.6.2.2.2.1.2

Rewrite the expression.

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

Step 3.6.2.2.2.2

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

Tap for more steps...

Step 3.6.2.2.2.2.1

Cancel the common factor.

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

Step 3.6.2.2.2.2.2

Divide $r$ by $1$ .

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

Step 3.6.2.2.3

Simplify the right side.

Tap for more steps...

Step 3.6.2.2.3.1

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

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

Step 3.6.2.2.3.2

Separate fractions.

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

Step 3.6.2.2.3.3

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

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

Step 3.6.2.2.3.4

Separate fractions.

$r = \frac{- 1}{- 2} \cdot \frac{1}{\cos (θ)} \tan (θ)$

Step 3.6.2.2.3.5

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

$r = \frac{- 1}{- 2} \sec (θ) \tan (θ)$

Step 3.6.2.2.3.6

Dividing two negative values results in a positive value.

$r = \frac{1}{2} \sec (θ) \tan (θ)$

Step 3.6.2.2.3.7

Combine $\frac{1}{2}$ and $\sec (θ)$ .

$r = \frac{\sec (θ)}{2} \tan (θ)$

Step 3.6.2.2.3.8

Combine $\frac{\sec (θ)}{2}$ and $\tan (θ)$ .

$r = \frac{\sec (θ) \tan (θ)}{2}$

Step 3.7

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

$r = 0$

$r = \frac{\sec (θ) \tan (θ)}{2}$

$r = 0$

$r = \frac{\sec (θ) \tan (θ)}{2}$