Algebra Examples

$x^{2} - y^{2} = 1$

Step 1

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

${(r \cos (θ))}^{2} - y^{2} = 1$

Step 2

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

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

Step 3

Solve for $r$ .

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify ${(r \cos (θ))}^{2} - {(r \sin (θ))}^{2}$ .

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Step 3.1.1.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = r \cos (θ)$ and $b = r \sin (θ)$ .

$(r \cos (θ) + r \sin (θ)) (r \cos (θ) - (r \sin (θ))) = 1$

Step 3.1.1.2

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

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Step 3.1.1.2.1

Apply the distributive property.

$r \cos (θ) (r \cos (θ) - r \sin (θ)) + r \sin (θ) (r \cos (θ) - r \sin (θ)) = 1$

Step 3.1.1.2.2

Apply the distributive property.

$r \cos (θ) (r \cos (θ)) + r \cos (θ) (- r \sin (θ)) + r \sin (θ) (r \cos (θ) - r \sin (θ)) = 1$

Step 3.1.1.2.3

Apply the distributive property.

$r \cos (θ) (r \cos (θ)) + r \cos (θ) (- r \sin (θ)) + r \sin (θ) (r \cos (θ)) + r \sin (θ) (- r \sin (θ)) = 1$

Step 3.1.1.3

Simplify terms.

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Step 3.1.1.3.1

Combine the opposite terms in $r \cos (θ) (r \cos (θ)) + r \cos (θ) (- r \sin (θ)) + r \sin (θ) (r \cos (θ)) + r \sin (θ) (- r \sin (θ))$ .

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Step 3.1.1.3.1.1

Reorder the factors in the terms $r \cos (θ) (- r \sin (θ))$ and $r \sin (θ) (r \cos (θ))$ .

$r \cos (θ) (r \cos (θ)) - r \cdot r \cos (θ) \sin (θ) + r \cdot r \cos (θ) \sin (θ) + r \sin (θ) (- r \sin (θ)) = 1$

Step 3.1.1.3.1.2

Add $- r \cdot r \cos (θ) \sin (θ)$ and $r \cdot r \cos (θ) \sin (θ)$ .

$r \cos (θ) (r \cos (θ)) + 0 + r \sin (θ) (- r \sin (θ)) = 1$

Step 3.1.1.3.1.3

Add $r \cos (θ) (r \cos (θ))$ and $0$ .

$r \cos (θ) (r \cos (θ)) + r \sin (θ) (- r \sin (θ)) = 1$

Step 3.1.1.3.2

Simplify each term.

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Step 3.1.1.3.2.1

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

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Step 3.1.1.3.2.1.1

Move $r$ .

$r \cdot r \cos (θ) \cos (θ) + r \sin (θ) (- r \sin (θ)) = 1$

Step 3.1.1.3.2.1.2

Multiply $r$ by $r$ .

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

Step 3.1.1.3.2.2

Multiply $r^{2} \cos (θ) \cos (θ)$ .

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Step 3.1.1.3.2.2.1

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

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

Step 3.1.1.3.2.2.2

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

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

Step 3.1.1.3.2.2.3

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

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

Step 3.1.1.3.2.2.4

Add $1$ and $1$ .

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

Step 3.1.1.3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.1.1.3.2.4

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

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Step 3.1.1.3.2.4.1

Move $r$ .

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

Step 3.1.1.3.2.4.2

Multiply $r$ by $r$ .

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

Step 3.1.1.3.2.5

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

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Step 3.1.1.3.2.5.1

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

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

Step 3.1.1.3.2.5.2

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

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

Step 3.1.1.3.2.5.3

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

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

Step 3.1.1.3.2.5.4

Add $1$ and $1$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Factor $r^{2}$ out of $r^{2} \cos^{2} (θ) + r^{2} (- 1 \sin^{2} (θ))$ .

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

Step 3.3

Factor.

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Step 3.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (θ)$ and $b = \sin (θ)$ .

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

Step 3.3.2

Remove unnecessary parentheses.

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $\cos (θ) + \sin (θ)$ .

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Step 3.4.2.1.1

Cancel the common factor.

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

Step 3.4.2.1.2

Rewrite the expression.

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

Step 3.4.2.2

Cancel the common factor of $\cos (θ) - \sin (θ)$ .

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Step 3.4.2.2.1

Cancel the common factor.

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

Step 3.4.2.2.2

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

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

Step 3.5

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

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

Step 3.6

Simplify $\pm \sqrt{\frac{1}{(\cos (θ) + \sin (θ)) (\cos (θ) - \sin (θ))}}$ .

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Step 3.6.1

Rewrite $\sqrt{\frac{1}{(\cos (θ) + \sin (θ)) (\cos (θ) - \sin (θ))}}$ as $\frac{\sqrt{1}}{\sqrt{(\cos (θ) + \sin (θ)) (\cos (θ) - \sin (θ))}}$ .

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

Step 3.6.2

Any root of $1$ is $1$ .

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

Step 3.6.3

Multiply $\frac{1}{\sqrt{(\cos (θ) + \sin (θ)) (\cos (θ) - \sin (θ))}}$ by $\frac{\sqrt{(\cos (θ) + \sin (θ)) (\cos (θ) - \sin (θ))}}{\sqrt{(\cos (θ) + \sin (θ)) (\cos (θ) - \sin (θ))}}$ .

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

Step 3.6.4

Combine and simplify the denominator.

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Step 3.6.4.1

$r = \pm \frac{\sqrt{(\cos (θ) + \sin (θ)) (\cos (θ) - \sin (θ))}}{\sqrt{(\cos (θ) + \sin (θ)) (\cos (θ) - \sin (θ))} \sqrt{(\cos (θ) + \sin (θ)) (\cos (θ) - \sin (θ))}}$

Step 3.6.4.2

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

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

Step 3.6.4.3

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

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

Step 3.6.4.4

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

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

Step 3.6.4.5

Add $1$ and $1$ .

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

Step 3.6.4.6

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

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Step 3.6.4.6.1

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

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

Step 3.6.4.6.2

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

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

Step 3.6.4.6.3

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

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

Step 3.6.4.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.6.4.1

Cancel the common factor.

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

Step 3.6.4.6.4.2

Rewrite the expression.

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

Step 3.6.4.6.5

Simplify.

$r = \pm \frac{\sqrt{(\cos (θ) + \sin (θ)) (\cos (θ) - \sin (θ))}}{(\cos (θ) + \sin (θ)) (\cos (θ) - \sin (θ))}$

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

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

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

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

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

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

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