Algebra Examples

$x^{2} + {(y - 3)}^{2} = 9$

Step 1

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

${(r \cos (θ))}^{2} + {(y - 3)}^{2} = 9$

Step 2

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

${(r \cos (θ))}^{2} + {((r \sin (θ)) - 3)}^{2} = 9$

Step 3

Solve for $r$ .

Tap for more steps...

Step 3.1

Subtract $9$ from both sides of the equation.

${(r \cos (θ))}^{2} + {((r \sin (θ)) - 3)}^{2} - 9 = 0$

Step 3.2

Simplify the left side of the equation.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

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

$r^{2} \cos^{2} (θ) + {((r \sin (θ)) - 3)}^{2} - 9 = 0$

Step 3.2.1.2

Rewrite ${(r \sin (θ) - 3)}^{2}$ as $(r \sin (θ) - 3) (r \sin (θ) - 3)$ .

$r^{2} \cos^{2} (θ) + (r \sin (θ) - 3) (r \sin (θ) - 3) - 9 = 0$

Step 3.2.1.3

Expand $(r \sin (θ) - 3) (r \sin (θ) - 3)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1.3.1

Apply the distributive property.

$r^{2} \cos^{2} (θ) + r \sin (θ) (r \sin (θ) - 3) - 3 (r \sin (θ) - 3) - 9 = 0$

Step 3.2.1.3.2

Apply the distributive property.

$r^{2} \cos^{2} (θ) + r \sin (θ) (r \sin (θ)) + r \sin (θ) \cdot - 3 - 3 (r \sin (θ) - 3) - 9 = 0$

Step 3.2.1.3.3

Apply the distributive property.

$r^{2} \cos^{2} (θ) + r \sin (θ) (r \sin (θ)) + r \sin (θ) \cdot - 3 - 3 (r \sin (θ)) - 3 \cdot - 3 - 9 = 0$

Step 3.2.1.4

Simplify and combine like terms.

Tap for more steps...

Step 3.2.1.4.1

Simplify each term.

Tap for more steps...

Step 3.2.1.4.1.1

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

Tap for more steps...

Step 3.2.1.4.1.1.1

Move $r$ .

$r^{2} \cos^{2} (θ) + r \cdot r \sin (θ) \sin (θ) + r \sin (θ) \cdot - 3 - 3 (r \sin (θ)) - 3 \cdot - 3 - 9 = 0$

Step 3.2.1.4.1.1.2

Multiply $r$ by $r$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin (θ) \sin (θ) + r \sin (θ) \cdot - 3 - 3 (r \sin (θ)) - 3 \cdot - 3 - 9 = 0$

Step 3.2.1.4.1.2

Multiply $r^{2} \sin (θ) \sin (θ)$ .

Tap for more steps...

Step 3.2.1.4.1.2.1

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

$r^{2} \cos^{2} (θ) + r^{2} (\sin (θ) \sin (θ)) + r \sin (θ) \cdot - 3 - 3 (r \sin (θ)) - 3 \cdot - 3 - 9 = 0$

Step 3.2.1.4.1.2.2

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

$r^{2} \cos^{2} (θ) + r^{2} (\sin (θ) \sin (θ)) + r \sin (θ) \cdot - 3 - 3 (r \sin (θ)) - 3 \cdot - 3 - 9 = 0$

Step 3.2.1.4.1.2.3

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

$r^{2} \cos^{2} (θ) + r^{2} {\sin (θ)}^{1 + 1} + r \sin (θ) \cdot - 3 - 3 (r \sin (θ)) - 3 \cdot - 3 - 9 = 0$

Step 3.2.1.4.1.2.4

Add $1$ and $1$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) + r \sin (θ) \cdot - 3 - 3 (r \sin (θ)) - 3 \cdot - 3 - 9 = 0$

Step 3.2.1.4.1.3

Move $- 3$ to the left of $r \sin (θ)$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 3 \cdot (r \sin (θ)) - 3 (r \sin (θ)) - 3 \cdot - 3 - 9 = 0$

Step 3.2.1.4.1.4

Multiply $- 3$ by $- 3$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 3 r \sin (θ) - 3 r \sin (θ) + 9 - 9 = 0$

Step 3.2.1.4.2

Subtract $3 r \sin (θ)$ from $- 3 r \sin (θ)$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 6 r \sin (θ) + 9 - 9 = 0$

Step 3.2.2

Simplify terms.

Tap for more steps...

Step 3.2.2.1

Combine the opposite terms in $r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 6 r \sin (θ) + 9 - 9$ .

Tap for more steps...

Step 3.2.2.1.1

Subtract $9$ from $9$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 6 r \sin (θ) + 0 = 0$

Step 3.2.2.1.2

Add $r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 6 r \sin (θ)$ and $0$ .

$r^{2} \cos^{2} (θ) + r^{2} \sin^{2} (θ) - 6 r \sin (θ) = 0$

Step 3.2.2.2

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

$r^{2} (\cos^{2} (θ)) + r^{2} \sin^{2} (θ) - 6 r \sin (θ) = 0$

Step 3.2.2.3

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

$r^{2} (\cos^{2} (θ)) + r^{2} (\sin^{2} (θ)) - 6 r \sin (θ) = 0$

Step 3.2.2.4

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

$r^{2} (\cos^{2} (θ) + \sin^{2} (θ)) - 6 r \sin (θ) = 0$

Step 3.2.3

Rearrange terms.

$r^{2} (\sin^{2} (θ) + \cos^{2} (θ)) - 6 r \sin (θ) = 0$

Step 3.2.4

Apply pythagorean identity.

$r^{2} \cdot 1 - 6 r \sin (θ) = 0$

Step 3.2.5

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

$r^{2} - 6 r \sin (θ) = 0$

Step 3.3

Factor $r$ out of $r^{2} - 6 r \sin (θ)$ .

Tap for more steps...

Step 3.3.1

Factor $r$ out of $r^{2}$ .

$r \cdot r - 6 r \sin (θ) = 0$

Step 3.3.2

Factor $r$ out of $- 6 r \sin (θ)$ .

$r \cdot r + r (- 6 \sin (θ)) = 0$

Step 3.3.3

Factor $r$ out of $r \cdot r + r (- 6 \sin (θ))$ .

$r (r - 6 \sin (θ)) = 0$

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 $0$ .

$r = 0$

$r - 6 \sin (θ) = 0$

Step 3.5

Set $r$ equal to $0$ .

$r = 0$

Step 3.6

Set $r - 6 \sin (θ)$ equal to $0$ and solve for $r$ .

Tap for more steps...

Step 3.6.1

Set $r - 6 \sin (θ)$ equal to $0$ .

$r - 6 \sin (θ) = 0$

Step 3.6.2

Add $6 \sin (θ)$ to both sides of the equation.

$r = 6 \sin (θ)$

Step 3.7

The final solution is all the values that make $r (r - 6 \sin (θ)) = 0$ true.

$r = 0$

$r = 6 \sin (θ)$

$r = 0$

$r = 6 \sin (θ)$