Convert to Polar x^2+4y^2=4

$x^{2} + 4 y^{2} = 4$

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

${(r \cos (θ))}^{2} + 4 y^{2} = 4$

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

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

Solve for $r$ .

Simplify the left side.

Simplify each term.

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

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

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

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

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

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

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

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

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

Divide each term in $r^{2} (\cos^{2} (θ) + 4 \sin^{2} (θ)) = 4$ by $\cos^{2} (θ) + 4 \sin^{2} (θ)$ and simplify.

Divide each term in $r^{2} (\cos^{2} (θ) + 4 \sin^{2} (θ)) = 4$ by $\cos^{2} (θ) + 4 \sin^{2} (θ)$ .

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

Simplify the left side.

Cancel the common factor of $\cos^{2} (θ) + 4 \sin^{2} (θ)$ .

Cancel the common factor.

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

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

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

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

$r = \pm \sqrt{\frac{4}{\cos^{2} (θ) + 4 \sin^{2} (θ)}}$

Simplify $\pm \sqrt{\frac{4}{\cos^{2} (θ) + 4 \sin^{2} (θ)}}$ .

Rewrite $\sqrt{\frac{4}{\cos^{2} (θ) + 4 \sin^{2} (θ)}}$ as $\frac{\sqrt{4}}{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}$ .

$r = \pm \frac{\sqrt{4}}{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}$

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

$r = \pm \frac{\sqrt{2^{2}}}{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}$

Pull terms out from under the radical, assuming positive real numbers.

$r = \pm \frac{2}{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}$

Multiply $\frac{2}{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}$ by $\frac{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}$ .

$r = \pm \frac{2}{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}} \cdot \frac{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}$

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}$ by $\frac{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}$ .

$r = \pm \frac{2 \sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)} \sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}$

Raise $\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}$ to the power of $1$ .

$r = \pm \frac{2 \sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}{{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}^{1} \sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}$

Raise $\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}$ to the power of $1$ .

$r = \pm \frac{2 \sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}{{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}^{1} {\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}^{1}}$

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

$r = \pm \frac{2 \sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}{{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}^{1 + 1}}$

Add $1$ and $1$ .

$r = \pm \frac{2 \sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}{{\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}^{2}}$

Rewrite ${\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}^{2}$ as $\cos^{2} (θ) + 4 \sin^{2} (θ)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}$ as ${(\cos^{2} (θ) + 4 \sin^{2} (θ))}^{\frac{1}{2}}$ .

$r = \pm \frac{2 \sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}{{({(\cos^{2} (θ) + 4 \sin^{2} (θ))}^{\frac{1}{2}})}^{2}}$

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

$r = \pm \frac{2 \sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}{{(\cos^{2} (θ) + 4 \sin^{2} (θ))}^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$r = \pm \frac{2 \sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}{{(\cos^{2} (θ) + 4 \sin^{2} (θ))}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$r = \pm \frac{2 \sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}{{(\cos^{2} (θ) + 4 \sin^{2} (θ))}^{\frac{2}{2}}}$

Rewrite the expression.

$r = \pm \frac{2 \sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}{{(\cos^{2} (θ) + 4 \sin^{2} (θ))}^{1}}$

Simplify.

$r = \pm \frac{2 \sqrt{\cos^{2} (θ) + 4 \sin^{2} (θ)}}{\cos^{2} (θ) + 4 \sin^{2} (θ)}$

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.

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

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

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

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

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

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

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

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

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

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

Convert to Polar x^2+4y^2=4

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