Convert the two rectangular (Cartesian) points to polar coordinates with $r > 0 $and$0 ≤ θ < 2π.$

Question

(a) Convert the two rectangular (Cartesian) points to polar coordinates with $r > 0 $and$0 ≤ θ < 2π.$

$(4,-4)$ and $(-1,\sqrt 3)$

(b) Convert the two polar points to rectangular (Cartesian) coordinates.

$(-1,\frac{\pi}{2})$ and $(6,-\frac{5\pi}{4})$

(c) Sketch the set of points $\left\{ (r,\theta ):1\le r\le 4,\frac { \pi }{ 4 } \le \theta \le \frac { 3\pi }{ 4 } \right\} $

My attempt

a) for $(4,-4)$ = $(r,\theta)$= $(4\sqrt 2,-\frac{\pi}{4})$

for $(-1,\sqrt 3)$ = $(r,\theta)$= $( 2,-\frac{\pi}{3})$

b)for $(-1,\frac{\pi}{2})$ = $(x,y)$ = $(0,1)$

for $(6,-\frac{5\pi}{4})$ = $(x,y)$ = $(-3\sqrt 3,-3\sqrt 3)$

Please verify my answers and also explain me how to do c)

Answer 1

For that

b)for $(-1,\frac{\pi}{2})$ = $(x,y)$ = $(0,1)$

it is strange that for r a negative value is given, but it should be $(0,-1)$.

For that

for $(6,-\frac{5\pi}{4})$ = $(x,y)$ = $(-3\sqrt 3,-3\sqrt 3)$

it should be $y>0$ and also $6\neq(54)^\frac12$.

To plot the region for part "c" let consider the boundaries: the two circles centered at the origin with radius $1$ and $4$ and the rays from the origin for $\theta=\pi/4$ and $\theta=3\pi/4$.

See for example for $2\le r\le 3$ and $\frac76\pi \le \theta \le \frac43 \pi$

Answer 2

You want to change your $-\frac{\pi}{4}$ in $$(4\sqrt 2,-\frac{\pi}{4})$$ to an angle between $0$ and $2\pi $

Similarly for the $$( 2,-\frac{\pi}{3})$$