Answer:
[tex]r^2 sin2\theta = 12[/tex]
Step-by-step explanation:
Given
[tex]xy = 6[/tex]
Required
Get polar equivalent
Polar coordinate [tex](r, \theta)[/tex] and rectangular coordinate [tex](x,y)[/tex] are related in such a way that:
[tex]x = rcos\theta[/tex] and [tex]y = rsin\theta[/tex]
So, substitute [tex]x = rcos\theta[/tex] and [tex]y = rsin\theta[/tex] in [tex]xy = 6[/tex]
[tex]rcos\theta * rsin\theta = 6[/tex]
This can be rewritten as:
[tex]r * cos\theta * r * sin\theta = 6[/tex]
Reorder
[tex]r * r * cos\theta * sin\theta = 6[/tex]
[tex]r^2 * cos\theta * sin\theta = 6[/tex]
Multiply both sides by 2
[tex]2 * r^2 * cos\theta * sin\theta = 6 * 2[/tex]
[tex]2 * r^2 * cos\theta * sin\theta = 12[/tex]
Reorder
[tex]r^2 * 2 * cos\theta * sin\theta = 12[/tex]
[tex]r^2 * 2cos\theta sin\theta = 12[/tex]
In trigonometry:
[tex]sin2\theta = 2sin\theta cos\theta = 2cos\theta sin\theta[/tex]
So, the expression becomes
[tex]r^2 * sin2\theta = 12[/tex]
[tex]r^2 sin2\theta = 12[/tex]
Hence, the equivalent of [tex]xy = 6[/tex] is [tex]r^2 sin2\theta = 12[/tex]
