Calculus Examples

$r^{2} = 2 \sin (2 θ)$

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.

$r^{2} = 2 (2 \sin (θ) \cos (θ))$

Step 1.1.2

Multiply $2$ by $2$ .

$r^{2} = 4 \sin (θ) \cos (θ)$

Step 2

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

$r^{2} = 4 \frac{y}{r} \cos (θ)$

Step 3

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

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

Step 4

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

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