Question

How do you convert `r = 2tanthetasectheta` into cartesian form?

Answer 1

By substitution `y = 1/2*x^2`

Explanation:

First convert the expression into terms of `cos(theta)` and `sin(theta)`
`r = 2*sin(theta)/cos(theta) *1/cos(theta)`
`rcos^2(theta) = 2*sin(theta)`
Now the normal substitutions are `x=rcos(theta)` and `y=rsin(theta)`
`:. sin(theta) = y/r` and `cos(theta) = x/r`

Substituting these values into the expression above gives
`r*(x/r)^2 = 2*y/r`
`cancel(r)*x^2/r^cancel(2) = 2*y/r`
`x^2 = r*2*y/r =2* y`
`:.y =1/2 x^2`