Question

Determine two pairs of polar coordinates for the point (4, -4) with 0° ≤ θ < 360°

Answer 1

x=4,y=-4
let \[4=r~\cos \theta,and ~-4=r \sin \theta \]
square and add
\[r^2(\cos ^2\theta+\sin ^2\theta )=4^2+(-4)^2=32\]
\[r=\pm \sqrt{32}=\pm 4\sqrt{2}\]
\[\sin \theta ~is~ negative ~and \cos \theta~is~positive.\]
so theta lies in 4th quadrant.
when \[r=4\sqrt{2},\cos \theta=\frac{ 4 }{ 4\sqrt{2} }=\frac{ 1 }{ \sqrt{2} }=\cos \left( 2 \pi-\frac{ \pi }{ 6 } \right)\]
\[\theta=\frac{ 11 \pi }{ 6 }\]
so polar co-ordinates are \[(4 \sqrt{2},\frac{ 11 \pi }{ 6 })\]
when \[r=-4\sqrt{2 },find \theta\]