Convert the point to rectangular coordinates. Show all work and give exact values (no decimals).

Question 1

Question 2

Given:

$(-4,(7\pi)/(6))$
$\text{Here r=-4 and }\theta=(7\pi)/(6)\text{.}$

Recall the formula for rectangular coordinates

$x=r\cos \theta$
$y=r\sin \theta$

$\text{Substitute r=-4 and }\theta=(7\pi)/(6)\text{ in x=rcos}\theta,\text{ we get}$

$x=-4\cos ((7\pi)/(6))$

$Use\cos ((7\pi)/(6))=\cos (\pi+(\pi)/(6))=-\cos ((\pi)/(6))=-\frac{\sqrt[]{3}}{2}\text{.}$

$x=-4(\frac{-\sqrt[]{3}}{2})=2\sqrt[]{3}_{}$

$\text{Substitute r=-4 and }\theta=(7\pi)/(6)\text{ in y=rsin}\theta,\text{ we get}$

$y=-4\sin ((7\pi)/(6))$

$Use\text{ sin}((7\pi)/(6))=sin(\pi+(\pi)/(6))=-sin((\pi)/(6))=-(1)/(2)\text{.}$

Hence the rectangular coordinates are

$(x,y)=(2\sqrt[]{3},\text{ 2)}$

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