Algebra Examples

$r = 2 \sin (3 θ)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $2 \sin (3 θ)$ .

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Step 1.1.1

Apply the sine triple-angle identity.

$r = 2 (- 4 \sin^{3} (θ) + 3 \sin (θ))$

Step 1.1.2

Apply the distributive property.

$r = 2 (- 4 \sin^{3} (θ)) + 2 (3 \sin (θ))$

Step 1.1.3

Multiply.

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Step 1.1.3.1

Multiply $- 4$ by $2$ .

$r = - 8 \sin^{3} (θ) + 2 (3 \sin (θ))$

Step 1.1.3.2

Multiply $3$ by $2$ .

$r = - 8 \sin^{3} (θ) + 6 \sin (θ)$

Step 2

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

$r = - 8 {(\frac{y}{r})}^{3} + 6 \frac{y}{r}$

Step 3

Multiply both sides by $r$ .

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Step 3.1

Multiply both sides by $r$ .

$r \cdot r = r (- 8 {(\frac{y}{r})}^{3} + 6 \frac{y}{r})$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply $r$ by $r$ .

$r^{2} = r (- 8 {(\frac{y}{r})}^{3} + 6 \frac{y}{r})$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify $r (- 8 {(\frac{y}{r})}^{3} + 6 \frac{y}{r})$ .

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Step 3.3.1.1

Simplify each term.

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Step 3.3.1.1.1

Apply the product rule to $\frac{y}{r}$ .

$r^{2} = r (- 8 \frac{y^{3}}{r^{3}} + 6 \frac{y}{r})$

Step 3.3.1.1.2

Combine $- 8$ and $\frac{y^{3}}{r^{3}}$ .

$r^{2} = r (\frac{- 8 y^{3}}{r^{3}} + 6 \frac{y}{r})$

Step 3.3.1.1.3

Move the negative in front of the fraction.

$r^{2} = r (- \frac{8 y^{3}}{r^{3}} + 6 \frac{y}{r})$

Step 3.3.1.1.4

Combine $6$ and $\frac{y}{r}$ .

$r^{2} = r (- \frac{8 y^{3}}{r^{3}} + \frac{6 y}{r})$

Step 3.3.1.2

Simplify terms.

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Step 3.3.1.2.1

Apply the distributive property.

$r^{2} = r (- \frac{8 y^{3}}{r^{3}}) + r \frac{6 y}{r}$

Step 3.3.1.2.2

Rewrite using the commutative property of multiplication.

$r^{2} = - r \frac{8 y^{3}}{r^{3}} + r \frac{6 y}{r}$

Step 3.3.1.2.3

Cancel the common factor of $r$ .

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Step 3.3.1.2.3.1

Cancel the common factor.

$r^{2} = - r \frac{8 y^{3}}{r^{3}} + r \frac{6 y}{r}$

Step 3.3.1.2.3.2

Rewrite the expression.

$r^{2} = - r \frac{8 y^{3}}{r^{3}} + 6 y$

Step 3.3.1.2.4

Cancel the common factor of $r$ .

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Step 3.3.1.2.4.1

Factor $r$ out of $- r$ .

$r^{2} = r \cdot - 1 \frac{8 y^{3}}{r^{3}} + 6 y$

Step 3.3.1.2.4.2

Factor $r$ out of $r^{3}$ .

$r^{2} = r \cdot - 1 \frac{8 y^{3}}{r \cdot r^{2}} + 6 y$

Step 3.3.1.2.4.3

Cancel the common factor.

$r^{2} = r \cdot - 1 \frac{8 y^{3}}{r \cdot r^{2}} + 6 y$

Step 3.3.1.2.4.4

Rewrite the expression.

$r^{2} = - \frac{8 y^{3}}{r^{2}} + 6 y$

Step 4

Since $r^{2} = x^{2} + y^{2}$ , replace $r^{2}$ with $x^{2} + y^{2}$ and $r$ with $\sqrt{x^{2} + y^{2}}$ .

$x^{2} + y^{2} = - \frac{8 y^{3}}{x^{2} + y^{2}} + 6 y$