Polar Coordinates — Question 7

Question 7

Problem

Find the Cartesian equation corresponding to $\theta=2\pi/3$ and explain why $r$ must be unrestricted.

Question 7 – Solution

Solution

1. Use the polar–Cartesian relationships $$x=r\cos\theta,\qquad y=r\sin\theta,\qquad r^2=x^2+y^2.$$ Equivalent polar coordinates satisfy $$(r,\theta)=(r,\theta+2k\pi)=(-r,\theta+(2k+1)\pi).$$

2. Apply the identity that matches the requested conversion, symmetry test, or intersection, and then check the resulting point or curve in the original polar equation.

3. The line has slope $\tan(2\pi/3)=-\sqrt3$, so $y=-\sqrt3x$.

4. Allowing negative $r$ supplies the opposite ray and therefore the entire line.