What is the Cartesian form of r^2-rtheta = -cottheta+3tantheta r2−rθ=−cotθ+3tanθ?

1 Answer
Shiva Prakash M V
Feb 18, 2018

Cartesian form of
r^2-rtheta=-cottheta+3tanthetar2−rθ=−cotθ+3tanθ is

sqrt(x^2+y^2)(sqrt(x^2+y^2)-tan^-1(y/x))=-x/y+(3y)/x√x2+y2(√x2+y2−tan−1(yx))=−xy+3yx

Explanation:

Given:
r^2-rtheta=-cottheta+3tanthetar2−rθ=−cotθ+3tanθ

x=rcosthetax=rcosθ
y=rsinthetay=rsinθ
r=sqrt(x^2+y^2)r=√x2+y2

theta=tan^-1(y/a)θ=tan−1(ya)
tantheta=y/xtanθ=yx

cottheta=x/ycotθ=xy

With this transformations,

r^2-rtheta=r(r-theta)r2−rθ=r(r−θ)
sqrt(x^2+y^2)(sqrt(x^2+y^2)-tan^-1(y/x))√x2+y2(√x2+y2−tan−1(yx))

-cottheta+3tantheta=-x/y+3y/x−cotθ+3tanθ=−xy+3yx

Thus,

sqrt(x^2+y^2)(sqrt(x^2+y^2)-tan^-1(y/x))=-x/y+(3y)/x√x2+y2(√x2+y2−tan−1(yx))=−xy+3yx

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