Answer:
Step-by-step explanation:
Convert rectangle (x , y) to polar coordinates ( r , θ)
[tex]x=r \cos \theta, y= r \sin \theta[/tex]
[tex]r=\sqrt{x^2+y^2} , \theta =tan^-^1 (\frac{y}{x} )[/tex]
a) converts (9, 0) to polar coordinates ( r , θ)
[tex]r=\sqrt{x^2+y^2} \\\\=\sqrt{9^2+0} \\\\=9[/tex]
[tex]\theta= \tan^-^1 (\frac{0}{9} )\\\\=0[/tex]
b) Convert [tex](18,\frac{18}{\sqrt{3} } )[/tex] to polar coordinates ( r, θ)
[tex]r = \sqrt{18^2+(\frac{18}{\sqrt{3} })^2 } \\\\=\sqrt{324+108} \\\\=\sqrt{432}[/tex]
[tex]\frac{x}{y} \theta = \tan^-^1(\frac{\frac{18}{\sqrt{3} } }{18} )\\\\= \tan ^-^1(\frac{1}{\sqrt{3} } )\\\\= \frac{\pi}{6}[/tex]
c) converts (-5, 5) to polar coordinates ( r , θ)
[tex]r =\sqrt{(-5)^2+(5)^2} \\\\=\sqrt{50} \\\\=5\sqrt{2}[/tex]
[tex]\theta=\tan^-^1(\frac{5}{-5} )\\\\= \tan^-^1(-1)\\\\=\frac{3\pi}{4}[/tex]
d) converts (-1, √3) to polar coordinates ( r , θ)
[tex]r=\sqrt{(-1)^2+(\sqrt{3})^2 } \\\\= \sqrt{4} \\\\=2[/tex]
[tex]\theta=\tan^-^1(\frac{\sqrt{3} }{-1} )\\\=\tan^-^1(-\sqrt{3} )\\\\=\frac{2\pi}{3}[/tex]
[tex]= \frac{2\pi}{\sqrt{3} }[/tex]
