Answer
(a) Ponit in Cartesian coordinates is $(-2, 2\sqrt 3)$
The point $(4, \dfrac{2\pi}{3})$ can be shown on polar graph as depicted.
(b) $$r=3\sqrt 2$$ and $$ \theta =3\pi/4$$
Work Step by Step
(a) from the graph it can be seen that the point $(4, 2\pi/3)$ is lie on a polar graph.
Cartesian coordinates are shown as:
$$x=rcos\theta=4 cos (2\pi/3)=-2$$
and $$y=rsin\theta=4 sin (2\pi/3)=2\sqrt 3$$
Points in Cartesian coordinates is: $(-2, 2\sqrt 3)$
(b) As we have $r=\sqrt {x^2+y^2}=3\sqrt 2$
$$x=rcos\theta=4 cos (2\pi/3)=-2$$ and
$$y=rsin\theta=4 sin (2\pi/3)=2\sqrt 3$$
Thus,
$-2=3\sqrt 2cos\theta$
$cos\theta=-\frac{1}{\sqrt 2}$ and $sin\theta=\frac{1}{\sqrt 2}$
As sine is positive and cosine is negative , and the angle $\theta$ lies in second quadrant.
$$ \implies \theta =3\pi/4$$
Hence, the result is: $r=3\sqrt 2$ and $ \theta =3\pi/4$
