Question

Convert the Cartesian equation x^4 - y^4 = 16 to a polar equation? r^2 = 16/cos2θ r^4 = 16/cos2θ r^2 = 8

Answer 1

Using the fact that
`x = r \cos \theta` and
`y = r \sin \theta`, we can write the Cartesian equation as


`x^4 - y^4 = 16`

`(r \cos \theta)^4 - (r \sin \theta)^4 = 16`

`r^4 (\cos^4 \theta - \sin^4 \theta) = 16`

`r^4 (\cos^2 \theta + \sin^2 \theta) (\cos^2 \theta - \sin^2 \theta) = 16`

Since
`\sin^2 \theta + \cos^2 \theta = 1` and
`\cos 2\theta = \cos^2 \theta - \sin^2 \theta`,


`r^4 (1) (\cos 2\theta) = 16`

`\bf r^4 = (16)/(\cos 2\theta)`