Algebra Examples

$x y = x + y$

Step 1

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

$(r \cos (θ)) \cdot y = (r \cos (θ)) + y$

Step 2

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

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

Step 3

Solve for $r$ .

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Step 3.1

Move all the expressions to the left side of the equation.

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Step 3.1.1

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

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

Step 3.1.2

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

$(r \cos (θ)) \cdot (r \sin (θ)) - r \cos (θ) - r \sin (θ) = 0$

Step 3.2

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

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Step 3.2.1

Move $r$ .

$r \cdot r \cos (θ) \cdot \sin (θ) - r \cos (θ) - r \sin (θ) = 0$

Step 3.2.2

Multiply $r$ by $r$ .

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

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

Step 3.3

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

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

Factor $r$ out of $- r \cos (θ)$ .

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

Step 3.3.1.3

Factor $r$ out of $- r \sin (θ)$ .

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

Step 3.3.1.4

Factor $r$ out of $r (r \cos (θ) \sin (θ)) + r (- 1 \cos (θ))$ .

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

Step 3.3.1.5

Factor $r$ out of $r (r \cos (θ) \sin (θ) - 1 \cos (θ)) + r (- 1 \sin (θ))$ .

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

Step 3.3.2

Rewrite $- 1 \cos (θ)$ as $- \cos (θ)$ .

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

Step 3.3.3

Rewrite $- 1 \sin (θ)$ as $- \sin (θ)$ .

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

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 $0$ .

$r = 0$

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

Step 3.5

Set $r$ equal to $0$ .

$r = 0$

Step 3.6

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

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Step 3.6.1

Set $r \cos (θ) \sin (θ) - \cos (θ) - \sin (θ)$ equal to $0$ .

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

Step 3.6.2

Solve $r \cos (θ) \sin (θ) - \cos (θ) - \sin (θ) = 0$ for $r$ .

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Step 3.6.2.1

Move all terms not containing $r$ to the right side of the equation.

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Step 3.6.2.1.1

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

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

Step 3.6.2.1.2

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

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

Step 3.6.2.2

Divide each term in $r \cos (θ) \sin (θ) = \cos (θ) + \sin (θ)$ by $\cos (θ) \sin (θ)$ and simplify.

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Step 3.6.2.2.1

Divide each term in $r \cos (θ) \sin (θ) = \cos (θ) + \sin (θ)$ by $\cos (θ) \sin (θ)$ .

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

Step 3.6.2.2.2

Simplify the left side.

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Step 3.6.2.2.2.1

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

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Step 3.6.2.2.2.1.1

Cancel the common factor.

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

Step 3.6.2.2.2.1.2

Rewrite the expression.

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

Step 3.6.2.2.2.2

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

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Step 3.6.2.2.2.2.1

Cancel the common factor.

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

Step 3.6.2.2.2.2.2

Divide $r$ by $1$ .

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

Step 3.6.2.2.3

Simplify the right side.

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Step 3.6.2.2.3.1

Simplify each term.

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Step 3.6.2.2.3.1.1

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

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Step 3.6.2.2.3.1.1.1

Cancel the common factor.

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

Step 3.6.2.2.3.1.1.2

Rewrite the expression.

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

Step 3.6.2.2.3.1.2

Convert from $\frac{1}{\sin (θ)}$ to $\csc (θ)$ .

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

Step 3.6.2.2.3.1.3

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

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Step 3.6.2.2.3.1.3.1

Cancel the common factor.

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

Step 3.6.2.2.3.1.3.2

Rewrite the expression.

$r = \csc (θ) + \frac{1}{\cos (θ)}$

Step 3.6.2.2.3.1.4

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

$r = \csc (θ) + \sec (θ)$

Step 3.7

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

$r = 0$

$r = \csc (θ) + \sec (θ)$

$r = 0$

$r = \csc (θ) + \sec (θ)$