Answer
(a) The point in Cartesian coordinates is $(-2, 2\sqrt 3)$
The point $(4, 2\pi/3)$ is shown on a polar graph.
(b) $r=3\sqrt 2$ and $ \theta =3\pi/4$
or: $r=-3\sqrt 2$ and $ \theta =7\pi/4$
Work Step by Step
(a) The point $(4, 2\pi/3)$ is shown on a polar graph.
The Cartesian coordinates are given by:
$x=rcos\theta=4 cos (2\pi/3)=-2$
$y=rsin\theta=4 sin (2\pi/3)=2\sqrt 3$
The point in Cartesian coordinates is $(-2, 2\sqrt 3)$
(b) $r=\sqrt {x^2+y^2}=3\sqrt 2$
$x=rcos\theta=4 cos (2\pi/3)=-2$
$y=rsin\theta=4 sin (2\pi/3)=2\sqrt 3$
Therefore,
$-2=3\sqrt 2cos\theta$
$cos\theta=-\frac{1}{\sqrt 2}$
$sin\theta=\frac{1}{\sqrt 2}$
Since sine is positive and cosine is negative, $\theta$ is in the 2nd quadrant.
and $\theta =3\pi/4$
Hence, $r=3\sqrt 2$ and $ \theta =3\pi/4$
