Question

Identify the curve by finding a cartesian equation for the curve r^2 sin 2 theta=1

Answer 1

The answer is
`2xy = 1`

The transformation between polar and cartesian coordinates is

`r = √(x^2+y^2),\qquad r\sin(\theta)=y,\qquad r\cos(\theta)=x`

Now use the trigonometric identity
`\sin(2\theta)=2\sin(\theta)\cos(\theta)` to rewrite the expression as

`r^2\sin(2\theta) = 2r^2\sin(\theta)\cos(\theta) = 2r\sin(\theta)r\cos(\theta)`

Now recall the transformations to rewrite

`2r\sin(\theta)r\cos(\theta) = 2xy`