Step-by-step explanation:
The relationship between rectangular form and polar form is given as follows:-
[tex]x=r*cos\theta, \;and\; y=r*sin\theta[/tex]
Part A: x=2
[tex]x=2\\r*cos\theta=2\\r=2/cos\theta\\r=2sec\theta[/tex]
Part B: x² + y² = 36
[tex]x^2+y^2=36\\(r*cos\theta)^2+(r*sin\theta)^2=36\\r^2*cos^2\theta+r^2*sin^2\theta=36\\r^2*(cos^2\theta+sin^2\theta)=36\\r^2*(1)=36\\r^2=36\\r=6[/tex]
Part C: x² + y² = 2y
[tex]x^2+y^2=2y\\(r*cos\theta)^2+(r*sin\theta)^2=2(r*sin\theta)\\r^2*(cos^2\theta+sin^2\theta)=2*r*sin\theta\\r^2=2*r*sin\theta\\r=2sin\theta[/tex]
Part D: x = √3 y
[tex]x = \sqrt{3} y \\ r*cos\theta=\sqrt{3}*r*sin\theta\\ cos\theta/sin\theta=\sqrt{3}\\cot\theta=\sqrt{3}\\\theta=cot^{-1}(\sqrt{3})\\\theta=\pi/6[/tex]
Part E: x = y
[tex]x=y\\r*cos\theta=r*sin\theta\\cos\theta=sin\theta\\sin\theta/cos\theta=1\\tan\theta=1\\\theta=tan^{-1}(1)\\\theta=\pi/4[/tex]
