Question

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

Answer 1

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

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((x)/(r)\right)\left((y)/(r)\right)`

Multiply:

`\displaystyle r = (4xy)/(r^2)`

Recall that x² + y² = r². Hence:

`\displaystyle r = (4xy)/(x^2 + y^2)`

By squaring both sides:

`\displaystyle r^2 = (16x^2y^2)/((x^2+y^2)^2)`

Substitute:

`\displaystyle x^2+y^2 = (16x^2y^2)/((x^2+y^2)^2)`

And multiply. Therefore:

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