Question

Convert (xy)^2=9 to an equation in polar coordinates

Answer 1

In polar coordinates, the equation
`\((xy)^2 = 9\)` becomes
`\(r^2 (1 - \cos(4t)) = 72\)`.

To convert the rectangular equation
`\((xy)^2 = 9\)` to polar coordinates, we use the relationships
`\(x = r \cos(t)\) and \(y = r \sin(t)\)`. Substitute these into the given equation:

`\[(xy)^2 = 9\]\[(r \cos(t) \cdot r \sin(t))^2 = 9\]\[r^2 \cos^2(t) \sin^2(t) = 9\]`

Now, use the trigonometric identity
`\(\cos^2(t) \sin^2(t) = (1)/(4) \sin^2(2t)\)`:

`\[r^2 \cdot (1)/(4) \sin^2(2t) = 9\]`

Multiply both sides by 4 to simplify:

`\[r^2 \sin^2(2t) = 36\]`

Now, using the double-angle identity
`\(\sin^2(2t) = (1)/(2)(1 - \cos(4t))\)`:

`\[r^2 \cdot (1)/(2)(1 - \cos(4t)) = 36\]`

Multiply both sides by 2 to clear the fraction:

`\[r^2 (1 - \cos(4t)) = 72\]`

Now, rearrange the equation to get it in the form
`\(r^n\)`:

`\[r^2 - r^2 \cos(4t) = 72\]\[r^2 (1 - \cos(4t)) = 72\]`

Therefore, the polar form of the given equation is
`\(r^2 (1 - \cos(4t)) = 72\)`.