Question

Convert r=2sin(2theta) into rectangular cords.

Answer 1

Answer:

$(x^2+y^2)^3 = 16x^2y^2$

Step-by-step explanation:

We want to convert the polar equation:

$\displaystyle r = 2 \sin 2\theta$

To rectangular form.

Recall the double-angle identity for sine:

$\displaystyle \sin 2\theta = 2\sin\theta\cos\theta$

Hence:

$\displaystyle r = 4\sin\theta\cos\theta$

Since x = rcosθ and y = rsinθ:

$\displaystyle r = 4\left(\frac{x}{r}\right)\left(\frac{y}{r}\right)$

Multiply:

$\displaystyle r = \frac{4xy}{r^2}$

Recall that x² + y² = r². Hence:

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

By squaring both sides:

$\displaystyle r^2 = \frac{16x^2y^2}{(x^2+y^2)^2}$

Substitute:

$\displaystyle x^2+y^2 = \frac{16x^2y^2}{(x^2+y^2)^2}$

And multiply. Therefore:

$(x^2+y^2)^3 = 16x^2y^2$