Question

Find the polar equivalent of xy=6

Answer 1

`r^2 sin2\theta = 12`

Explanation:

Given

`xy = 6`

Required

Get polar equivalent

Polar coordinate
`(r, \theta)` and rectangular coordinate
`(x,y)` are related in such a way that:

`x = rcos\theta` and
`y = rsin\theta`

So, substitute
`x = rcos\theta` and
`y = rsin\theta` in
`xy = 6`

`rcos\theta * rsin\theta = 6`

This can be rewritten as:

`r * cos\theta * r * sin\theta = 6`

Reorder

`r * r * cos\theta * sin\theta = 6`

`r^2 * cos\theta * sin\theta = 6`

Multiply both sides by 2

`2 * r^2 * cos\theta * sin\theta = 6 * 2`

`2 * r^2 * cos\theta * sin\theta = 12`

Reorder

`r^2 * 2 * cos\theta * sin\theta = 12`

`r^2 * 2cos\theta sin\theta = 12`

In trigonometry:

`sin2\theta = 2sin\theta cos\theta = 2cos\theta sin\theta`

So, the expression becomes

`r^2 * sin2\theta = 12`

`r^2 sin2\theta = 12`

Hence, the equivalent of
`xy = 6` is
`r^2 sin2\theta = 12`