Algebra Examples

$r = 3 \cos (2 θ)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $3 \cos (2 θ)$ .

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Step 1.1.1

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

$r = 3 (2 \cos^{2} (θ) - 1)$

Step 1.1.2

Apply the distributive property.

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

Step 1.1.3

Multiply.

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Step 1.1.3.1

Multiply $2$ by $3$ .

$r = 6 \cos^{2} (θ) + 3 \cdot - 1$

Step 1.1.3.2

Multiply $3$ by $- 1$ .

$r = 6 \cos^{2} (θ) - 3$

Step 2

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

$r = 6 {(\frac{x}{r})}^{2} - 3$

Step 3

Multiply both sides by $r$ .

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Step 3.1

Multiply both sides by $r$ .

$r \cdot r = r (6 {(\frac{x}{r})}^{2} - 3)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply $r$ by $r$ .

$r^{2} = r (6 {(\frac{x}{r})}^{2} - 3)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify $r (6 {(\frac{x}{r})}^{2} - 3)$ .

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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 $\frac{x}{r}$ .

$r^{2} = r (6 \frac{x^{2}}{r^{2}} - 3)$

Step 3.3.1.1.2

Combine $6$ and $\frac{x^{2}}{r^{2}}$ .

$r^{2} = r (\frac{6 x^{2}}{r^{2}} - 3)$

Step 3.3.1.2

Simplify terms.

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Step 3.3.1.2.1

Apply the distributive property.

$r^{2} = r \frac{6 x^{2}}{r^{2}} + r \cdot - 3$

Step 3.3.1.2.2

Cancel the common factor of $r$ .

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Step 3.3.1.2.2.1

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

$r^{2} = r \frac{6 x^{2}}{r \cdot r} + r \cdot - 3$

Step 3.3.1.2.2.2

Cancel the common factor.

$r^{2} = r \frac{6 x^{2}}{r \cdot r} + r \cdot - 3$

Step 3.3.1.2.2.3

Rewrite the expression.

$r^{2} = \frac{6 x^{2}}{r} + r \cdot - 3$

Step 3.3.1.2.3

Move $- 3$ to the left of $r$ .

$r^{2} = \frac{6 x^{2}}{r} - 3 r$

Step 4

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

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