Algebra Examples

$r = 2 \sin (2 θ)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $2 \sin (2 θ)$ .

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Step 1.1.1

Apply the sine double-angle identity.

$r = 2 (2 \sin (θ) \cos (θ))$

Step 1.1.2

Multiply $2$ by $2$ .

$r = 4 \sin (θ) \cos (θ)$

Step 2

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

$r = 4 \frac{y}{r} \cos (θ)$

Step 3

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

$r = 4 \frac{y}{r} \frac{x}{r}$

Step 4

Multiply both sides by $r$ .

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Step 4.1

Multiply both sides by $r$ .

$r \cdot r = r (4 \frac{y}{r} \frac{x}{r})$

Step 4.2

Simplify the left side.

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Step 4.2.1

Multiply $r$ by $r$ .

$r^{2} = r (4 \frac{y}{r} \frac{x}{r})$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify $r (4 \frac{y}{r} \frac{x}{r})$ .

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Step 4.3.1.1

Rewrite using the commutative property of multiplication.

$r^{2} = 4 r (\frac{y}{r} \cdot \frac{x}{r})$

Step 4.3.1.2

Multiply $\frac{y}{r} \cdot \frac{x}{r}$ .

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Step 4.3.1.2.1

Multiply $\frac{y}{r}$ by $\frac{x}{r}$ .

$r^{2} = 4 r \frac{y x}{r \cdot r}$

Step 4.3.1.2.2

Raise $r$ to the power of $1$ .

$r^{2} = 4 r \frac{y x}{r^{1} r}$

Step 4.3.1.2.3

Raise $r$ to the power of $1$ .

$r^{2} = 4 r \frac{y x}{r^{1} r^{1}}$

Step 4.3.1.2.4

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

$r^{2} = 4 r \frac{y x}{r^{1 + 1}}$

Step 4.3.1.2.5

Add $1$ and $1$ .

$r^{2} = 4 r \frac{y x}{r^{2}}$

Step 4.3.1.3

Cancel the common factor of $r$ .

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Step 4.3.1.3.1

Factor $r$ out of $4 r$ .

$r^{2} = r \cdot 4 \frac{y x}{r^{2}}$

Step 4.3.1.3.2

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

$r^{2} = r \cdot 4 \frac{y x}{r \cdot r}$

Step 4.3.1.3.3

Cancel the common factor.

$r^{2} = r \cdot 4 \frac{y x}{r \cdot r}$

Step 4.3.1.3.4

Rewrite the expression.

$r^{2} = 4 \frac{y x}{r}$

Step 4.3.1.4

Combine $4$ and $\frac{y x}{r}$ .

$r^{2} = \frac{4 y x}{r}$

Step 5

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

$x^{2} + y^{2} = \frac{4 y x}{\sqrt{x^{2} + y^{2}}}$